UC-NRLF 


SB    Eh 


LIVE-LOAD    STRESSES 


IN 


RAILWAY  BRIDGES 

WITH 

FORMULAS  AND  TABLES 


BY 

GEORGE  E.  BEGGS,  A.B.,  C.E. 

Assistant  Professor  of  Civil  Engineering  in  Princeton  University; 
Associate  Member  of  the  American  Society  of  Civil  Engineers; 
Member  of  the  Society  for  the  Promotion  of  Engineering  Education 


FIRST  EDITION 
FIRST    THOUSAND 


NEW  YORK 
JOHN  WILEY  &  SONS,  INC. 

LONDON  :    CHAPMAN    &    HALL,   LIMITED 
1916 


IV  PREFACE 

The  author  wishes  to  acknowledge  his  indebtedness  to 
the  American  Bridge  Company  for  material  assistance,  and 
in  particular  to  Mr.  0.  E.  Hovey,  Assistant  Chief  Engineer 
of  this  company,  for  his  encouragement  and  help.  The 
author  also  desires  to  acknowledge  the  valuable  suggestions 
made  in  the  revision  of  the  original  text  by  Professor  F.  H. 
Constant,  of  the  Civil  Engineering  Department  of  Princeton. 
To  Professor  William  H.  Burr  of  Columbia  University,  the 
writer  is  permanently  indebted  for  the  logical  and  thorough 
instruction  received  from  him  as  a  student. 

G.  E.  B. 

PRINCETON  UNIVERSITY 
December,  1915. 


CONTENTS 


ARTICLE  I. 
ARTICLE       II. 

ARTICLE  III. 

ARTICLE  IV. 

ARTICLE  V. 

ARTICLE  VI. 

ARTICLE  VII. 

ARTICLE  VIII. 

ARTICLE  IX. 

ARTICLE  X. 


Influence  Lines.     Definition  and  Use 


PAGE 

1 


Sum  and  Rate  of  Variation  of  Ordinate-load  Products 
between  Two  Diverging  Lines 

Sum  and  Rate  of  Variation  of  Ordinate-load  Products 
for  any  Influence  Line.  Position  of  Loading  for 
Maximum  Live-load  Stress 

Girder  Bridge  without  Panels.  General  Formulas 
for  Reactions,  Shears,  and  Bending  Moment  with 
its  Rate  of  Variation 

General  Formulas  for  Pier  Reaction  and  its  Rate  of 
Variation.  Illustrative  Problem 

Girder  Bridge  with  Panels.  General  Formulas  for 
Live-load  Stresses  and  their  Rate  of  Variation. 
Illustrative  Problem  .  . 

Through  Pratt  Truss.  General  Formulas  for  Live- 
load  Stresses  and  their  Rate  of  Variation.  Illus- 
trative Problems 

Three-hinged  Arch.  Application  of  the  General 
Method  to  the  Calculation  of  Live-load  Stresses 


13 


23 


27 


31 


48 


Equivalent  Uniform  Loads 54 

Method  of  Calculating  Table  of  Load  Sums  and 
Moment  Sums  for  any  Standard  Loading.  Illus- 
trative Example 57 


ARTICLE      XI.     Summary  of  Formulas 


59 


Tables  1  to  21 


67 


LIVE-LOAD   STRESSES 

ARTICLE  I. 

INFLUENCE   LINES.      DEFINITION  AND   USES. 

INFLUENCE  lines  are  useful  in  determining  the  position 
of  live  load  on  a  bridge  to  produce  maximum  effect.  They 
offer  also  a  convenient  method  of  deriving  general  alge- 
braic formulas  for  stresses  and  rules  for  maximum  when 
the  general  relations  between  influence  lines  and  algebraic 
formulas  are  once  understood;  and  in  the  case  of  the  more 
complex  problems  of  skew  bridges,  arches,  cantilever  bridges, 
etc.,  the  influence  lines  themselves  serve  as  a  most  direct 
method  for  the  determination  of  the  maximum  live-load 
stresses. 

An  influence  line  may  be  defined  as  a  line  showing  the 
variation  in  any  function  caused  by  a  single  unit  load  as 
it  moves  across  the  bridge.  Vertical  loads  only  will  be 
considered.  The  function  may  be  a  reaction,  bending 
moment,  shear,  stress,  deflection,  or  any  quantity  what- 
soever at  a  given  part  of  a  bridge,  provided  that  its  value 
is  a  function  of  the  position  of  the  unit  load  on  the  bridge. 

Refer  to  Fig.  la.  Consider  the  span  AB,  and  let  Z  be 
any  function  at  the  fixed  position  C  on  the  span  L.  If 
the  load  unity  moves  across  the  span  AB  and  the  value 
of  Z  be  calculated  for  each  position  of  the  unit  load  and 
its  value  z  plotted  below  the  corresponding  position  of  this 
load  as  an  ordinate  from  a  horizontal  base  line,  the  locus 
of  the  plotted  points  will  be  the  influence  line  for  Z.  For 
example,  if  Z  be  the  bending  moment  at  the  fixed  section 
C  in  a  beam  of  span  L,  the  influence  line  will  be  as  shown 
in  Fig.  Ib.  In  plotting  influence  lines,  ordinates  repre- 

l 


2  LP'E-LOAD    STRESSES 

sen  ting  positive  quantities  are  plotted  above  the  base  line; 
and  negative,  below.  In  case  the  influence  line  consists  of 
several  straight  segments,  it  is  necessary  to  determine  the 
value  of  the  ordinates  only  where  the  influence  line  has  a 
change  of  direction;  i.e.,  at  the  salient  points.  For  example, 


FIG    1. 

the  points  A,  C,  and  B  are  the  salient  points  of  the  influence 
line  in  Fig.  Ib. 

The  value  of  Z  caused  by  a  single  load  w  is  equal  to 
wz,  if  z  is  the  influence  ordinate  below  w.  The  value  of  Z 
caused  by  a  series  of  loads  w^  wZj  w3,  etc.,  is 

.     .     (1) 


LIVE-LOAD    STRESSES  3 

where  Zi,  z2,  z3,  etc.,  are  the  influence  ordinates  below  the 
corresponding  loads.  It  will  be  convenient  to  speak  of  such 
a  quantity  as  wz  as  an  ordinate-load  product. 

Formula  (1)  therefore  may  be  expressed  thus: 

Z  =  Sum  of  ordinate-load  products. 

The  area  between  the  influence  line  and  the  base  line 
is  called  the  influence  area.  It  may  be  shown  that  the 
value  of  Z  caused  by  a  uniform  load  on  the  bridge  is  pro- 
portional to  the  area  Az  of  the  influence  line  between  the 
ordinates  at  the  extremities  of  the  uniform  load.  If  the 
uniform  load  in  Fig.  la  has  an  intensity  of  q  per  unit  of 
length,  the  load  in  the  length  dx  equals  q  dx,  and  the  influ- 
ence of  this  elementary  load  on  the  value  of  Z  is  zq  dx, 
where  z  is  the  influence  ordinate  below  q  dx.  Summing  up 
for  the  length  of  the  uniform  load, 


z  = 


If  a  series  of  equal  loads  w  is  on  the  span,  the  value  of 
Zis 

Z  =  zwz  =  w2z  ........    (3) 


If  a  series  of  unequal  loads,  w\,  w2,  etc.,  is  multiplied  by 
the  corresponding  ordinates  of  an  influence  line  or  a  por- 
tion of  an  influence  line  which  has  a  constant  ordinate  z, 
as  in  Fig.  Ic,  the  value  of  Z  is 


Z  =  z(wi  +  w2  +...)=  zSw  =  zW     •    •    •    (4) 

where  W  equals  the  sum  of  these  loads. 

If  a  series  of  unequal  loads  is  multiplied  by  the  corre- 
sponding ordinates  of  an  influence  line  or  a  portion  of  an 
influence  line  consisting  of  two  diverging  lines,  as  shown 
in  Fig.  Id,  the  value  of  Z,  or  the  sum  of  the  ordinate  load 
products,  and  the  rate  at  which  Z  varies  as  the  loading 
advances,  are  given  by  the  two  theorems  that  follow.  The 
slope  of  a  line  is  defined  at  the  beginning  of  Art.  2. 


4  LIVE-LOAD    STRESSES 

Theorem  I. 

The  sum  of  the  ordinate-load  products  between  two  di- 
verging lines  equals  the  difference  between  the  slopes  of 
the  two  lines  multiplied  by  the  sum  of  the  moments  of 
the  loads  about  the  intersection  of  these  lines. 

In  symbols,  this  is  stated  as 

Z  =  CaMa  V  .......   (5) 

* 

Theorem  II. 

The  rate  at  which  the  sum  of  the  -ordinate-load  prod- 
ucts between  the  two  diverging  lines  increases  as  the  load- 
ing moves  away  from  the  intersection  of  these  lines  equals 
the  difference  between  the  slopes  of  the  two  lines  multiplied 
by  the  sum  of  the  loads. 

In  symbols,  this  is  stated  as 

dZ=  «CJU_  CM,  (ga) 

dx  dx  °  dx 

The  proofs  of  these  theorems  follow  in  the  next  article. 


-.  ARTICLE  II. 

SUM  AND   RATE   OF  VARIATION   OF   ORDINATE-LOAD   PRODUCTS 
BETWEEN   THE    TWO    DIVERGING    LINES. 

CONSIDER  the  diverging  lines  DAB  and  AC  in  Fig.  2. 
Use  the  following  notation: 
w  =  any  vertical  load. 
z  =  ordinate  below  w  in  the  angle  BAC. 
Z  =  2wnzn  =  sum  of  ordinate-load  products. 


Salient  Point 


FIG.  2. 

Ma  =  2wnxn  =  moment  sum  of  all  loads  to  left  of  A  a 

about  A. 

Wa  =  2wn  =  load  sum  of  all  loads  to  left  of  A  a. 
SR  =  slope  of  line  DA  =  tangent  of  angle  which  DA 

makes  with  the  horizontal. 


6  LIVE-LOAD    STRESSES 

SL  =  slope  of  line  AC  =  tangent   of  angle  which  AC 

makes  with  the  horizontal. 

Ca  =  —  —  (SL  —  SR)  =  length  of  ordinate  unit  distance 
xn 

from  A. 

Slopes  are  counted  numerically  positive  when  upward 
to  the  left.  The  sign  of  Ca  (called  the  coefficient  at  salient 
point  A)  is,  accordingly,  negative  when  AC  diverges  below 
DA  produced  to  the  left  of  A.  The  value  of  Ca  may  be 

determined  graphically  as  --  or  it  may  be  figured  algebra- 

xn 

ically  as  (SL  —  SR). 

Proof  of  Theorem  I,  or  that  Z  =  CaMa. 

Consider  the  load  wn  distant  xn  from  the  salient  point  a. 
By  the  similar  triangles  AEF  and  AGH, 

Ca 


1.00         Xn9 

Therefore, 

wnzn  =  Cawnxn (A) 

Summing  up  all  of  the  ordinate-load  products, 

Z  =  2wnzn  =  Ca2wnxn  =  CaMa (5) 

Proof  of  Theorem  II,  or  that  -^  =  CaWa. 

From  equation  (A)  above,  the  increase  in  the  ordinate- 
load  product  wnzn  for  an  advance  dxn  of  the  load  is 

wndzn  =  Ca-wn-dxn. 

Summing  up  the  increases  of  all  the  ordinate-load  products 
and  noting  that  dx  is  the  same  for  all  loads, 

dZ  =  2wndzn  =  Cadx-2wn  =  Ca-Wa-dx. 
Dividing  by  dx,  -r-  =  CaWa  =  — | 


ARTICLE   III. 


SUM  AND   RATE   OF  VARIATION   OF  ORDINATE-LOAD   PRODUCTS 

FOR  ANY   INFLUENCE   LINE.       POSITION   OF   LOADING   FOR 

MAXIMUM    LIVE-LOAD    STRESS. 

AN  influence  line  of  a  general  type  is  shown  in  Fig.  3, 
this  one  in  t  particular  being  for  the  member  U^L*  of  the 


Salient 


Internals 


C)rflinates 


Slopes 


Coefficients 


Influence  Line  for  U3  L4 

of 
189'  Three  Hinged  Arch 


Moment  Sums 


t-f 


T- 


FIG.  3. 


arch  shown  in  Fig.  15.  It  is  assumed  that  the  ordinates  at 
all  salient  points  and  the  intervals  between  these  points  are 
known.  Ordinates  and  slopes  are  counted  positive  or  nega- 
tive as  already  defined.  The  slope  of  any  segment  of  the 

7 


8  LIVE-LOAD    STRESSES 

influence  line  equals  the  ordinate  at  the  left  minus  the 
ordinate  at  the  right  end  of  this  segment  divided  by  the 
corresponding  interval.  The  coefficient  C  at  any  salient 
point  equals  the  slope  of  the  segment  at  the  left  minus  the 
slope  of  the  segment  at  the  right  of  this  point.  The  sub- 
tractions in  each  case  are  made  algebraically. 

It  should  be  remembered,  as  has  already  been  pointed 
out  in  Art.  2,  that  the  value  of  any  coefficient  C  may  also 
be  measured  graphically  from  an  influence  line  which  has 
been  drawn  to  scale.  For  example,  in  Fig.  3  the  value  of 

the  coefficient  C2  =  —  and  C4  =  - 

Xi  X4 

The  algebraic  calculation  of  the  coefficients  at  all  sa- 
lient points  of  the  influence  line  in  Fig.  3  is  given  below. 
If  it  be  assumed  that  this  influence  line  has  been  drawn 
to  scale,  the  signs  of  the  numerical  values  of  the  slopes  and 
coefficients  will  be  as  given  in  the  parentheses. 

«i  =  °-*2   (+)  Ci  =  0  -  *,     (-) 


S2  =    "~7~  *   (  —  )  C2  =  Si  —  S2       (  +  ) 


4  =   S3   -  S4 

Cs  =  s4  -  0      (+) 

A  numerical  evaluation  of  the  slopes  and  coefficients  for 
this  influence  line  is  given  in  Fig.  15  of  Art.  8,  which  the 
reader  should  check  in  order  to  understand  completely  the 
method  of  procedure.  These  coefficients  should  also  be 
checked  by  the  graphical  method  as  already  explained. 

2  59 
For  example,  in  Fig.  15  the  value  of  C2  =  -~-  =  .0863. 

It  will  be  noted  in  the  algebraic  calculation  of  the  coeffi- 
cients C  at  all  salient  points  that  each  slope  enters  once 


LIVE-LOAD    STRESSES  9 

as  positive  and  once  as  negative.     Therefore  the  sum  of  all 
coefficients  equals  zero. 

SC  =  0  .....    .....    (6) 

This  formula  serves  as  a  check  on  the  values  of  the  coef- 
ficients which  have  been  determined  either  by  calculation 
or  by  graphical  measurement. 

The  general  formulas  for  the  sum  of  the  ordinate-load 
products  for  any  influence  line  (viz.,  with  several  salient 
points  such  as  the  one  shown  in  Fig.  3)  may  be  arrived 
at  by  considering  the  two  contiguous  sloping  sides  of  the 
influence  line  meeting  at  each  salient  point  as  two  diverg- 
ing lines.  The  entire  influence  line  is  thus  made  up  of  pairs 
of  diverging  lines  (see  Fig.-  3)  to  each  pair  of  which  for- 
mula (5)  may  be  directly  applied.  Thus  in  Fig.  3, 

Ordinate-load  products  in  \dfc  =  CiMi  (  —  ) 

"  [c0e  =  C2M2  (+) 

"  \eha  =  C3M3  (-) 

"  =  CM,  (+) 


(+) 


The  signs  of  the  CM's  are  +  or  —  according  to  the 
signs  of  the  coefficients,  for  the  M'  s  are  always  positive. 
Summing  up  the  above  equations  and  observing  that  the 
ordinate-load  products  cancel  one  another  except  between  the 
influence  line  fghkm  and  its  base  line  fom,  it  follows  that 
the  sum  of  the  ordinate-load  products  for  the  influence 
line,  or  the  live-load  stress,  is 

S  =  dM^  +  C2MZ  +  .  .  .  =  ZCM.  ....   (7) 

The  letter  S  represents  in  general  any  stress  or  sum  of 
ordinate-load  products  for  any  influence  line,  while  Z  stands 
for  the  sum  of  ordinate-load  products  for  any  geometrical 
figure. 

The  rate  at  which  S  varies  as  the  load  advances  a  dis- 
tance dx  equals 


10  LIVE-LOAD    STRESSES 

dS 


, 

---  -j  -----  h 
c  c  a# 

But  by  formula  (5a)  this  becomes 

=  dTFi  +  C2TF2  +  .  .  .  =  SCT7.    ;,v..   (8) 


Wi,  W2,  etc.,  =  sum  of  all  of  the  loads  to  the  left  of 
points  1,  2,  etc.,  respectively,  whether  on  the  span  or  not. 

Mi,  Mz,  etc.,  =  moment  of  the  same  loads  about  points 
1,  2,  etc.,  respectively,  whether  on  the  span  or  not. 

The  above  formulas  (6),  (7),  and  (8)  apply  equally  well 
when  the  loading  is  headed  from  left  to  right  instead  of 
from  right  to  left,  the  latter  being  the  more  usual  way. 
In  applying  these  formulas,  however,  it  will  save  confu- 
sion not  to  reverse  the  loading,  but  to  turn  the  influence 
line  end  for  end,  for  this  operation  changes  neither  the 
values  nor  the  signs  of  the  coefficients  C. 

The  stress  S  —  2CM  is  related  to  its  derivative  -  --  = 

ax 

2CTP  in  the  same  way  that  any  function  is  related  to  its 

i  Q 

derivative.    Thus,  if  the  value  of  -j-  passes  through  zero  as 

the  loading  advances,  the  stress  itself  may  have  reached 
any  one  of  four  conditions;  namely, 

1.  Numerically  maximum  positive  value. 

2.  "  minimum        "  " 

3.  "  maximum  negative    " 

4.  minimum 

In  practice  it  is  desirable  to  find  the  positions  of  load- 
ing to  satisfy  the  first  and  third  conditions.  This  may  be 
done  by  proceeding  as  directed  below.  It  is  assumed  in 
stating  the  following  rules  that  the  live  load  is  advancing 
from  right  to  left.  In  case  the  live  load  advances  from 
left  to  right,  the  wheel  will  be  tried  first  to  the  left  and 


LIVE-LOAD    STRESSES  11 

then  to  the  right  of  a  salient  point.  In  other  words,  dx 
is  always  an  increment  in  the  same  direction  as  the  loading 
advances. 

Rule  1. — To  determine  the  position  of  loading  to  give 
a  maximum  positive  stress,  place  the  live  load  on  the  part 
of  the  bridge  corresponding  to  the  positive  portion  of  the 
influence  line.  Try  a  wheel  first  immediately  to  the  right 
of  a  salient  point  that  has  a  negative  coefficient  and  then 

just  to  the  left  of  this  point.     Calculate  the  value  of  -7-  = 

ctx 

I,CW  for  each  of  these  successive  positions  of  loading.     If 

j  & 

the  sign  of  -7-  changes  from  +  to  — ,  a  position  of  load- 
ed 

ing  for  maximum  positive  stress  is  determined. 

Rule  2. — To  determine  the  position  of  loading  to  give 
a  numerically  maximum  negative  stress,  place  the  live  load 
on  that  part  of  the  bridge  corresponding  to  the  negative 
portion  of  the  influence  line.  Try  a  wheel  first  immedi- 
ately to  the  right  of  a  salient  point  that  has  a  positive  coef- 
ficient and  then  just  to  the  left  of  this  point.  Calculate 

7Q 

the  value  of  -5—  =  2CTF  for  each  of  these  successive  posi- 
tions of  loading.  If  the  sign  of  -r-  changes  from  —  to  +, 

a  position  of  loading  for  numerically  maximum  negative 
stress  is  determined. 

It  will  be  noted  that  the  negative  coefficients  C  occur 
at  those  salient  points  where  the  angles  of  the  influence 
line  point  upward,  while  the  positive  coefficients  C  occur 
at  those  salient  points  where  the  angles  point  downward. 

It  is  unnecessary  to  seek  a  position  of  loading  for  maxi- 
mum positive  stress  by  placing  a  wheel  successively  to  the 
right  and  to  the  left  of  any  salient  point  which  has  a  posi- 
tive coefficient;  for  if  ^  =  2(7TF  be  +  when  the  wheel  is 
to  the  right  of  this  point,  it  would  have  a  still  larger  + 


12  LIVE-LOAD    STRESSES 

value  when  the  wheel  is  to  the  left  of  the  point.     A  change, 

TO 

therefore,  of  -7-  from  +  to  —  would  not  result.     Similarly, 

it  may  be  shown  to  be  unnecessary  to  seek  a  numerically 
maximum  negative  stress  by  trying  wheels  at  any  salient 
point  which  has  a  negative  coefficient. 

Formulas  (7)  and  (8)  are  the  general  formulas  for  the 
solution  of  the  sum  of  the  ordinate-load  products  of  an 
influence  line  and  the  rate  of  change  of  this  sum,  and  are 
applicable  to  any  form  of  influence  line.  They  give  at  once 
a  definite  solution  of  the  position  of  a  set  of  loads  produc- 
ing maximum  positive  and  negative  stresses  in  any  member 
of  any  truss  or  girder  for  which  an  influence  line  can  be 
drawn  and  the  values  of  such  stresses.  The  method  is 
particularly  advantageous  in  the  case  of  statically  indeter- 
minate structures,  such  as  two-hinged  and  no-hinged  arches, 
swing  bridges,  continuous  girders,  etc.,  where  general  ana- 
lytical criteria  for  the  positions  of  loads  producing  maxi- 
mum stresses  cannot  readily  be  expressed  and  where  such 
maximum  stresses  have  had  to  be  found  by  assuming  posi- 
tions of  loadings  and  scaling  the  influence-line  ordinates 
under  all  the  loads,  a  laborious  process  and  one  open  to 
much  liability  of  mechanical  inaccuracy. 

In  applying  the  present  method  to  the  simple  forms  of 
girders  and  trusses  (viz.,  the  statically  determinate  struc- 
tures where  the  ordinates  of  the  influence  lines  are  readily 
expressible  algebraically)  it  will  generally  be  more  conve- 
nient to  transform  formulas  (7)  and  (8)  in  each  case  whereby 
the  coefficients  C  may  be  expressed  in  terms  of  the  geo- 
metric proportions  of  the  truss  or  girder.  This,  in  the  fol- 
lowing articles  (4  to  7  inclusive),  we  shall  proceed  to  do  for 
the  case  of  girder  bridges  (with  and  without  panels),  pier 
reactions,  and  through  Pratt  trusses  with  curved  or  hori- 
zontal chords.  The  general  method  will,  however,  be  ap- 
plied directly  to  the  case  of  the  three-hinged  arch  in  Art. 
8,  which  will  serve  as  a  typical  example  of  the  application 
of  the  method  to  any  influence  line. 


ARTICLE   IV. 

GIRDER   BRIDGE    WITHOUT   PANELS. 

IN  Fig.  4  is  shown  a  girder  bridge  without  panels.     The 
live  load  has  advanced  beyond  the  span,  this  being  the 


Point  (( l 


FlG.    4. 

most  general  case.  Formulas  for  the  end  reactions  and  for 
the  bending  moment  and  shear  at  any  section  will  be 
developed. 

13 


14  LIVE-LOAD    STRESSES 

The  influence  line  for  Ri  is  shown  in  Fig.  4a.  The  sum 
of  the  ordinate-load  products  within  the  shaded  area  rst 
equals  the  end  reaction  Ri,  which  at  the  same  time  is  the 
end  shear  at  Ri. 

From  Fig.  4a, 

Ordinate-load  products  in  \rst  = 

"   (ate 

—       "         "  "          "   [orb 

"   brsc. 
By  using  formulas  (4)  and  (5),  this  equation  becomes 

Rl  =       M3-lMl-Wl  =  ^-Wl.  .  (9) 


Any  value  of  M  or  W  may  be  read  directly  from  Table 
2  for  the  standard  loadings  given  in  Table  1.  For  example, 
in  Fig.  4,  if  h  =  10',  k  =  30',  and  wl  of  Cooper's  #50  has 
advanced  14'  beyond  the  left  end  of  the  span,  we  have 
from  Table  2, 

At  1,  14'  from  wi,  M1  =    350.0*1  W1  =    62.50* 

At  2,  24'  from  Wi,  M2  =  1150.0  Wz  =  112.50 

At  3,  54'  from  wi,  M,  =  5435.0  TF3  =  177.50 

The  formula  for  R2  is  developed  as  for  JBi,  the  method 
of  writing  the  second  member  of  the  first  equation  bein^ 
abbreviated  in  a  way  readily  understood.  From  the  influ- 
ence line  in  Fig.  4b,  and  the  formulas  (4)  and  (5), 

R2  =  Ordinate-load  products  in  (dvxe  —  \  dvf  -\-  \fue) 
Or 

R2  =  Ws  -  ±Mt  +  ~M,  =  W,  -  M*~Ml    .    (Qa) 

The  sum  of  the  reactions  Ri  and  R2  as  given  by  (9)  and 
(9a)  equals  TF3  —  W\,  or  the  sum  of  the  loads  on  the  bridge. 

From  the  influence  line  in  Fig.  4c  and  formulas  (5)  or 
(7),  the  equation  for  bending  moment  may  be  written: 

M  =  Ordinate-load  products  in  (  |  gbh  —    \  gak  +  \kz 


LIVE-LOAD    STRESSES  15 

Or 

M  =  lj-M,  +  |jf,  -  M,       ....   (10) 

Formula  (10)  readily  follows,  likewise,  from  the  general 
formula  (7),  S  =  C.M,  +  <72M2  +  C3M3  =  SCM. 

For  example,  in  the  case  of  the  bending  moment  at 
point  2  in  Fig.  4, 


C  k       ^  1 

'  L  ~  L  ~~ 

C  ,.•*      0 

cs-L- 

Whence  M  =  r  M,  -  M2  +  4  M3   .  (lOa) 


Taking  the  derivative  of  M  with  respect  to  the  advance 
dx  of  the  loading  toward  the  left  or  using  formula  (8) 
directly,  the  rate  of  variation  of  the  bending  moment  is 

fF.  +  -TF,       ....   (11) 


.  .  ,       ... 

All  positions  for  maximum  M  may  be  found  by  trying 
wheels  at  point  2  as  directed  by  Rule  1  of  Art.  3.  In  ap- 
plying this  rule  the  simultaneous  shifting  of  other  wheels 
of  the  rigid  loading  from  right  to  left  of  points  1  and  3 
as  a  wheel  is  shifted  from  right  to  left  of  point  2,  must  be 
taken  into  account  by  substituting  in  formula  (11)  the 
corresponding  changed  values  of  Wi  and  W3.  It  is  to  be 
remembered,  as  stated  in  Art  3,  that  it  is  entirely  unnec- 
essary to  try  wheels  at  points  1  and  3. 

From  the  influence  line  in  Fig.  4d,  the  formula  for  the 
intermediate  shear  S  follows  by  applying  formulas  (4) 
and  (5)  : 

S  =  Ordinate-load  products  in 

(|  mfq  —  mden  —  \  ncq) 


16  LIVE-LOAD    STRESSES 

Or 

S  =  j-M3  -  TT.  -  \M,  =  M*  ~  Mj  -  W>    .    (12) 

There  is  one  more  thing  to  be  borne  in  mind  in  calcula- 
ting maximum  bending  moments  in  a  girder  bridge  without 
panels:  it  is  the  rule  for  finding  the  section  where  the 
absolute  maximum  bending  moment  occurs.  The  rule  is  often 
spoken  of  as  the  "centre  of  gravity  rule"  and  may  be  stated 
as  follows: 

The  bending  moment  under  any  given  wheel  becomes  maxi- 
mum when  the  centre  of  the  span  bisects  the  distance  from  the 
wheel  in  question  to  the  centre  of  gravity  of  the  loading  on  the 
span. 

In  the  practical  application  of  this  rule,  the  procedure 
is  first  to  find  the  wheel  which  gives  maximum  bending 
moment  at  the  centre  of  the  span  and  then  to  shift  this 
wheel  so  that  the  bending  moment  beneath  it  becomes  an 
absolute  maximum  according  to  the  centre  of  gravity  rule. 
For  the  usual  standard  loadings  the  maximum  centre  mo- 
ment closely  approximates  the  absolute  maximum  bending 
moment  for  the  spans  greater  than  70  feet. 

The  proof  of  the  centre  of  gravity  rule  follows.  Refer 
to  Fig.  5.  Assume  that  it  has  been  found  by  trial  that 
the  wheel  wn  gives  the  maximum  centre  moment.  The 
general  case  where  load  has  advanced  beyond  the  span  is 
taken.  In  order  to  get  an  absolute  maximum  bending 
moment  under  wn,  this  wheel  must  be  shifted  a  certain 
distance  from  the  centre.  Let  such  position  be  distance  y 
from  Ri.  The  sum  of  the  loads  on  the  span  is  called  P2 
and  equals  (W3  —  Wi).  The  centre  of  gravity  of  the  loads 
P2  is  distance  x  from  R2.  The  sum  of  the  loads  on  the 
span  to  the  left  of  wn  is  called  Pi,  and  their  centre  of  gravity 
is  at  the  fixed  distance  b  from  wn. 

Taking  moments  about  R2, 


LIVE-LOAD    STRESSES 


17 


Therefore, 


P  r 

M  =  R,y  -  P,6  =  ~  y  - 


In  this  equation  for  M,  the  only  variables  are  x  and 
/y.  Therefore,  M  will  be  a  maximum  when  the  product 
xy  is  maximum.  Note,  however,  that  the  sum 

x  +  y  =  (L  —  a)  =  constant. 

If  two  variables  have  a  constant  sum,  their  product  is 
maximum  when  the  two  variables  are  equal.  Therefore,  M 
is  maximum  when  x  =  y.  But  when  x  =  y,  the  distance 
from  wn  to  the  centre  of  gravity  of  the  loading  is  bisected 


FIG.  5. 


by  the   centre   of   the  span.     This  proves  the   centre  of 
gravity  rule. 

In  order  to  apply  this  rule,  a  general  expression  for  x 
is  needed. 

Since  Ri  =  -y2-  it  follows  that  x  =  -5—'      Substitute  the 
LJ  r  2 

value  of  Ri  from  formula  (9),  and  the  value  (Ws  —  Wi)  for 
P2. 


M3  - 

x  '' 


!  -  LW  i 


-  Wl 


(13) 


18  LIVE-LOAD    STRESSES 

In  the  special  case  where  the  loading  has  not  advanced 
beyond  the  left  end  of  the  span,  M i  and  Wi  equal  zero  and 
x  becomes 

-       M3  , 

x~-:w,    (13a) 

Problems  relating  to  a  girder  bridge  without  panels  will 
now  be  given  to  illustrate  the  application  of  the  above 
formulas  and  the  use  of  some  of  the  tables  following  the 
text. 

Problem. — Given  a  40-foot  deck-girder  bridge  consisting 
of  one  girder  per  rail.  Use  Cooper's  E5Q  loading.  Find 
the  maximum  shear  at  the  end,  quarter  point,  and  centre. 
Determine  also  the  maximum  bending  moment  at  the 
quarter  point  and  at  the  centre,  and  the  absolute  maximum 
bending  moment.  All  values  are  to  be  given  per  rail. 

Solution. — Table  5  following  the  text  gives  the  position 
of  Cooper's  loadings  for  maximum  end  shear.  This  table 
is  the  result  of  the  solution  of  end  shears  for  a  large  num- 
ber of  spans.  As  a  general  rule,  however,  it  is  safe  to  as- 
sume that  w2  of  Cooper's  and  similar  loadings  will  always 
give  the  maximum  end  or  intermediate  shear  when  placed 
immediately  to  the  right  of  the  given  section,  the  live  load 
being  headed  toward  the  left.  The  exceptions  in  Table  5 
to  this  general  rule  are  not  of  prime  importance,  for  the 
actual  value  of  the  shear  when  w2  is  used  is  sufficiently  close 
to  the  maximum  even  in  the  exceptional  cases.  There  is 
no  satisfactory  criterion  for  determining  the  position  of  load- 
ing for  maximum  shear  in  girder  bridges  without  panels,  for 
it  is  as  easy  to  calculate  the  actual  values  of  the  shears  for 
the  successive  positions  of  loading  as  it  is  to  apply  any 
criterion.  In  the  case  of  bending  moment,  however,  time 
is  saved  by  using  the  criterion. 

Maximum  End  Shear. 
Use  formula  (9),  R,  =  ^~^  _  Wi.     Place    wheel   2 


LIVE-LOAD    STRESSES  19 

of  Cooper's  #50  immediately  to  right  of  Ri.  Take  the 
values  of  moment  and  load  sums  for  Cooper's  E5Q  from 
Table  2. 

Maximum  end  shear  =  :        ,         --  12.5  =  94.3*. 


Maximum  Shear  at  Quarter  Point. 
Use  formula  (12)  with  w2  at  quarter  point. 


t 

S  at  Y±  point  =  28384705  ~  °  -  12.5  =  58.5*. 

Maximum  Shear  at  Centre. 
Using  formula  (12)  with  w2  at  centre. 


at  centre  =  -  12.5  =  27.5*. 


The  values  for  the  shears  are  given  in  Kips,  or  thou- 
sand of  pounds.  A  comparison  of  the  above  shears  with 
those  in  Table  7  shows  agreement  of  results. 

Maximum  Bending  Moment  at  the  One-Quarter  Point. 
First  compute  successive  pairs  of  values  for-;—  for  dif- 

ferent wheels,  first  placed  to  the  right  and  then  to  the  left 
of  the  quarter  point.  A  change  of  sign  from  +  to  —  indi- 
cates a  wheel  that  gives  a  maximum.  Use  formula  (11), 

' 


at  J4  point. 

: :     ^f  =  M  (H2.5)  +  -M  (o)  -  o  =  + 

No  maximum. 


20  LIVE-LOAD   STEESSES 


^  =  M  (H2.5)  +  M  (0)  -  12.5  =  + 
wz  at  J4  point. 

~  =  V±  (145)  +  M  (0)  -  12.5  =  + 

Maximum. 
=  M  (145)  +  M  (0)  -  37.5  =  - 

waw 

at  J4  point. 


M  (145)  +  M  (12.5)  -  37.5  =  + 

Maximum. 

~  =  K  (161.25)  +  M  (12.5)  -  62.5  =  - 
at  J^  point. 

y±  (161.25)  +  %  (12.5)  -  62.5  =  - 

No  maximum. 
M  (177.5)  +  M  (37.5)  -  87.5  =  - 

Accordingly,  compute  the  value  of  M  by  formula  (10) 
for  wz  and  ws  at  quarter  point. 

M  =  l£Ms  +  jMi  -  M2 (10) 

M  for  w2  at  quarter  point, 

M  =  }/±  (2838.75)  +  %  (0)  -  100  =  609.7  Kip  feet. 

M  for  w3  at  quarter  point, 

M  =  K  (3563.75)  +  ^  (37.5)  -  287.5  =  631.6  Kip  feet. 

The  latter  value,  631.6,  is  the  maximum  bending  mo- 
ment at  the  quarter  point.     A  comparison  of  this  value 


LIVE-LOAD    STRESSES 


21 


with  Table  11  shows  agreement  of  results.  Reference  to 
Table  3  indicates  that  the  correct  wheel  for  maximum  has 
been  chosen. 

Maximum  Bending  Moment  at  the  Centre. 

dM      W3+  W1       TJ7    ,in  , 
-fa  = 2 W*>  '  Oa^' 


M 3  +  MI       ,..     /v*  \      i       h 

M  = M2,  (Ha),  when 


at  centre, 


dM      128.75 
dx  2 

128.75 


-  62.5  =  + 


^  at  centre, 


dM      145 


dM       145 


No  maximum. 


Maximum. 


at  centre, 


145  +  12.5 


dx  '  2 

161.25+  12.5 


dx 


-  87.5  =  - 

No  maximum. 

-  112.5  =  - 


Therefore,   maximum   centre   moment   occurs  with  w4   at 
centre. 

M  -  283^'75  -  600  =  819.37  Kip  feet. 

Zj 

This  value  agrees  with  Table  11;    and  the  position  of 
loading,  with  Table  3. 


22  LIVE-LOAD    STRESSES 

Absolute  Maximum  Bending  Moment. 

Shift  w4  according  to  centre  of  gravity  rule,  and  then 
recompute  the  value  of  M  under  this  wheel  by  formula  (10). 
Note  that  new  values  for  Zi,  Z2,  and  M3  must  be  determined. 

By  formula  (13a),  when  w*  is  at  the  centre, 

-  _  M,  _  2838.75  _. 

'  W3  '       145 

Therefore  for  absolute  maximum  bending  moment  under 
104,  shift  loading  to  left  -  — ^ —  -  =  0'.21. 

The  new  values  of  Zi,  Z2,  and  M3  are 

h  =  20.00  -  0.21  =  19.79 
Z2  =  20.00  +  0.21  =  20.21 

M3  =  2838.75  +  .21(145)  =  2869.2 

The  absolute  maximum  bending  moment  = 
M  =  j-  M3  -h  T  Mi  —  M2 

=  i^  (2869.2)  +  0  -  600  =  819.54  Kip  feet. 


It  appears,  therefore,  that  the  absolute  maximum  bend- 
ing moment  is  .17  Kip  feet  greater  than  the  maximum  cen- 
tre moment.  The  difference  is  not  great  in  this  particular 
case,  as  the  required  shift  of  the  loading  is  comparatively 
small.  The  position  of  loading  for  absolute  maximum  bend- 
ing moment  agrees  with  Table  4,  and  its  value  agrees  with 
Table  7. 


ARTICLE  V. 

PIER   REACTION. 

IN  Fig.  4e  is  given  the  influence  line  for  the  pier  reac- 
tion R  between  two  non-continuous  beam  spans  li  and  Z2. 
From  this  influence  line,  the  formulas  (5)  and  (7)  give 

R  =  Ordinate-load  products  in  (|  gbh  —  \  gak  +  \  kzh) 

Or, 

MX      Mi       L  L 

R-  T  +  T  -  UM2  = 

Formula  (14)  may  also  be  derived  from  formula  (10) 
since  the  ordinates  of  the  influence  line  for  R  bear  the  con- 

stant ratio  7-7  to  the  corresponding  influence  ordinates  for 
LI  li 

M,  the  position  of  the  live  load  and  the  values  of  li  and  12 
remaining  fixed. 
Therefore, 


R  =  ~ 


Substituting  the  value  M  =  j-  M  3  +  j-  Mi  —  M2  from 

formula  (10)  in  formula  (16),  the  result  is  again  formula 
(14). 

For  equal  spans, 

777  4-  t>      M3  +  MJ-  2M2 

li  =  Z2  =  I  so  that  R  =  -         —j—  .   (14a) 

The  rate  of  change  of  R  for  a  movement  dx  of  the 
loading  to  the  left  is 

dR     W,      Wl       L  L  /Z,  I 

~dx  =T  +  T  "  u*  w*  =  £E  (~LW*  +  Tw>  ~ 

23 


24  LIVE-LOAD    STRESSES 

For  equal  spans,  h  =  12  =  I,  so  that 

dR       Ws  +  W,  -  2W* 

dx  =  I 


(15a) 


In  the  last  member  of  formula  (15)  the  quantity  within 
the  parentheses  is  the  same  as  the  expression  for  -y—  in 

formula  (11).  It  follows,  therefore,  that  the  same  position 
of  loading  gives  maximum  R  and  maximum  M  for  any 
given  values  of  l\  and  £2- 

Problem.  —  (a)  Find  the  maximum  pier  reaction  per  rail 
between  two  simple  beam  spans  li  =  10  ft.  and  Z2  =  30  ft. 
(b)  Find  the  maximum  pier  reaction  between  two  simple 
beam  spans,  each  having  a  length  of  20  feet.  Use  Cooper's 
#50  loading. 

Solution  of  Problem  (a). 

Use  formula  (15)  to  find  position  of  loading  for 
maximum  R. 

•  -(15) 


at  pier. 

dR  40      /1<K          ,   30 


!  fi(0)  -12.6)- 


dx     '  10X30V40V  r40 

Maximum. 

dR  4°     S  (145) +g(0)- 37.5)  =  - 


__ 
da;     •  10  X  30iO  40 

t  at  pier. 
dR  40 


4       - 


Maximum. 

dR  40       (W  L?Pn2^       fi9 

"d^    -10^30  Uo(  '40° 

Use  formula  (14)  to  compute  the  value  of  R. 


LIVE-LOAD    STRESSES  25 

R__  M*.    Mi        L_M 

wz  at  pier. 

283875  0           40                         k 

30  "  10  "  10  X  30  ( 
w3  at  pier. 

_    3563.75  37.5           40                     _      t 

30  10"    "  10  X  30  (l 

The  latter  value  of  84fc  is  the  maximum  pier  reaction. 
Its  value  agrees  with  Table  14  and  the  position  of  loading 
agrees  with  Table  3. 

Solution  of  Problem  (b). 
Use  formulas  (14a)  and  (15a), 

R  =  --          —j—       — ,  and  —j—  =  -        -y-       — . 

w3  at  pier. 

A**.       128.75  +  0  -  2  X  37.5 
dx  20 

No  maximum. 
dR       128.75  +  0  -  2  x  62.5 


dx  20 

w4  at  pier. 

dR_  145  +  0  -  2  X  62.5 

dx   ''  20 

Maximum. 

dR_  145  +  0  -  2  x  87.5  _ 

dx   ''  20 

w$  at  pier. 

dR_  145  +  12.5  -  2  X  87.5  _ 

dx   ''  20 

No  maximum. 

dR_  161.25  +  12.5  -  2  X  112.5 

dx   '  20 

Therefore,  maximum  pier  reaction  occurs  when  w±  is  at  the 
pier. 


26  LIVE-LOAD    STRESSES 

2838.75  -  0  -  2  X  600  * 

R  -  -2Q-  51.9  . 

This  maximum  pier  reaction  of  81.9fc  agrees  with  value 
in  Table  7  and  Table  14,  while  the  position  of  loading  agrees 
with  that  given  by  Table  3. 


ARTICLE  VI. 

GIRDER   BRIDGE   WITH   PANELS. 

In  Fig.  6  is  shown  a  girder  bridge  with  panels.      It  is  as- 


FIG.  6. 

sumed  that  the  live  load  has  advanced  beyond  the  left  end 
of  the  span,  this  being  the  most  general  case. 

27 


28  LIVE-LOAD    STRESSES 

The  formulas  for  Ri  and  R2  are  the  same  as  formulas 
(9)  and  (9a)  for  the  girder  without  panels,  if  the  girder 
bridge  with  panels  has  end  floor-beams;  but  if  this  bridge 
has  end  struts  with  the  end  stringers  resting  on  separate 
pedestals,  the  value  of  Ri  beneath  the  end  of  the  main 
girder  is  the  same  as  Sa,  the  shear  in  the  end  panel,  as 
given  by  formula  (17)  to  follow. 

Inasmuch  as  the  maximum  bending  moment  in  a  beam 
carrying  concentrated  loads  always  occurs  beneath  a  con- 
centration, the  maximum  bending  moments  in  the  main 
girder  of  a  girder  bridge  with  panels  will  occur  at  the  floor- 
beams.  The  influence  line  for  the  bending  moment  at  the 
floor-beams  is  the  same  as  for  the  bending  moment  in  a 
girder  bridge  without  panels;  accordingly,  formulas  (10)  and 
(11)  are  to  be  used  in  finding  maximum  bending  moments 
at  the  floor-beams. 

It  remains  to  derive  formulas  for  the  maximum  shears 
Sa  in  the  end  panel  and  Sb  in  any  intermediate  panel.  In 
Fig.  6  are  given  the  influence  lines  for  Sa  and  Sb.  The 
correctness  of  the  ordinates  is  at  once  evident.  The  slopes 
and  coefficients  are  calculated  as  explained  in  Arts.  2  and  3. 
The  general  formulas  for  Sa  and  Sb  and  their  rates  of  varia- 
tion may  be  written  at  once  by  use  of  formulas  (7)  and  (8). 

8   =     M,  +       ^i  -     ^«  = 


T*—Z*-Z 

L  p  p 


(19) 
(20) 


Formula  (17)  when  compared  with  formula  (10)  shows 
that  Sa  is  equal  to  the  bending  moment  at  the  first  inter- 
mediate floor-beam  divided  by  the  length  of  the  first  panel. 
Formula  (18)  when  compared  with  formula  (11)  shows  that 


LIVE-LOAD    STRESSES  29 

the  same  position  of  loading  that  gives  maximum  bending 
moment  at  the  first  intermediate  floor-beam  will  also  give 
maximum  shear  in  the  end  panel. 

Formulas  (19)  and  (20)  are  perfectly  general  and  will 
serve  for  any  assumed  series  of  vertical  loads  in  any  posi- 
tion. For  the  usual  standard  loadings  and  panel  lengths, 
however,  it  is  not  necessary  to  advance  any  loads  beyond 
an  intermediate  panel  for  maximum  shear  in  this  panel. 
Therefore,  for  practical  purposes  formulas  (19a)  and  (20a) 


Illustrative  Problem.  —  A  single  track  through  girder 
bridge  with  a  floor  system  consisting  of  stringers  and  floor- 
beams,  both  end  and  intermediate,  has  six  panels  of  20  feet 
each.  Find  the  maximum  end  reaction  and  the  shear  in 
panels  0  —  1,1—2,  and  2  —  3,  using  Cooper's  E5Q  loading. 

Solution.  —  For  maximum  end  reaction  place  wheel  2  at 
left  end.  Use  formula 


(9) 


12.6  -  217.1* 


Note  that  the  above  value  agrees  with  Table  7. 
For  maximum  shear  in  panel  0  —  1,  find  critical  wheel 
by  formula  (18)  and  then  compute  shear  by  formula  (17). 
Try  wheel  3  at  panel  point  1. 


Maximum. 

H  •  %(£ (365>  -  °  - 62-5)  -  - 


30  LIVE  -LOAD    STRESSES 

Note  that  the  position  of  loading  agrees  with  Table  3. 
For  this  position  of  loading  formula  (17)  gives 

Sa  =  i(    (21895)  +  0  -  287.5)  M  168.1*. 


For  maximum  shears  in  the  intermediate  panels,  deter- 
mine the  position  of  loading  by  formula  (20a)  and  the  shear 
by  formula  (19a). 


(20a) 
(19a) 


Panel  1-2.     Try  wheel  3  at  panel  point  2. 

§  =  27)(-6  <306-25)  -  37'5)  -  + 

Maximum. 

f  =  1(J  (322.50)  -62.5)  =  - 

Sb  =  ^o(l  (1505L25)  "  287-5)  z~-  11L0*- 

Panel  2-3.     Try  wheel  3  at  panel  point  3. 


Maximum. 


&  =  ^Q  (9345)  -  287.5)  =  63.5*. 

The  above  values  for  shears  agree  with  the  values  given 
by  Table  9.  The  wheel  for  maximum  shear  in  panels  of 
girder  and  truss  bridges  is  given  in  Table  6. 


ARTICLE   VII. 


THROUGH     PRATT     TRUSS.       GENERAL     FORMULAS    FOR    LIVE- 
LOAD      STRESSES      AND      THEIR      RATE      OF      VARIATION. 
ILLUSTRATIVE    PROBLEMS. 

dS 

THE  general  formulas  S  =  2CM  and  -7—  =  2CW  may 

be  used  to  write  the  equations  for  the  live-load  stresses  in 
any  member  of  a  framed  structure  as  soon  as  its  influence 


Salient  Points 


Ordinates 


Slopes 


Coefficients 


mkp 


h 
nv 


JL 
nv 


FIG.  7. 

line  has  been  drawn  and  the  ordinates  at  the  salient  points 
determined. 

In  Figs.  7,  8,  9,  and  10  are  shown  all  the  influence  lines 

31 


32  LIVE-LOAD    STRESSES 

needed  in  writing  the  formulas  for  the  live-load  stresses  in 
a  through  Pratt  truss  with  non-parallel  or  parallel  chords. 
The  influence  ordinate  at  any  salient  point  is  the  calcu- 
lated stress  due  to  a  one-pound  load  on  the  bridge  at  the 
panel  point  above  this  salient  point.  By  easily  discovered 
relations  between  similar  triangles,  the  algebraic  value  of 
each  stress,  or  influence  ordinate,  is  expressed  in  terms  that 
are  most  readily  evaluated  in  any  numerical  problem. 

The  derivation  of  any  one  formula  for  a  live-load  stress 
is  typical.  Refer  to  Fig.  7.  The  stress  in  the  lower  chord 
member  S5  is  found  by  taking  moments  about  C.  The 
influence  line  for  $5  is  straight  over  each  of  the  two  inter- 
vals kp  and  mp.  The  ordinates  at  the  salient  points  1  and 
4  are  zero.  The  ordinate  at  salient  point  3  must  be  found 
by  placing  a  one-pound  load  at  the  lower  panel  point  of 
the  truss  above  this  salient  point  and  calculating  the  value 
of  S6.  For  the  unit  load  so  placed, 

Reaction  at  A  =  —  =  - 
np       n 

By  moments  about  C, 


Therefore, 


~  (mp)  =  S,  (v) 


=  +  —  ~  =  Influence  ordinate  at  3. 

nv 


The  slopes  of  the  segments  of  this  influence  line  follow. 

,  mkp  k 

Slope  of  ao  =  --     -  -r-  mp  =  -- 
nv  nv 

r  ,  .    mkp  m 

Slope  of  be  =  +  —  -  -r-  kp  =  -\  -- 

nv  nv 

The  coefficients  C  for  use  in  the  general  formula  S  = 
2CM  are  now  found. 

c1  =  o  +  -  =  +- 

nv  nv 


LIVE-LOAD    STRESSES  33 

—        —  -          - 
nv       nv  v 


nv  nv 

Therefore,  for  the  position  of  the  live  load  advanced 
beyond   the  limits   of  the   span,   the  general  formula  for 


However,  in  actual  practice  it  is  usually  not  necessary 
to  advance  the  loading  beyond  the  left  end  of  the  span  in 
order  to  get  a  maximum  value  of  S$.  The  usual  formula 
will  therefore  not  contain  the  term  Mi,  since  this  will  be 
zero;  thus, 


Inasmuch  as  the  horizontal  component  of  the  stress  *S8 
in  an  inclined  top  chord  member  or  end  post  equals  the 
stress  $5  in  a  corresponding  lower  chord  member,  the  stress 
$6  in  any  top  chord  member  or  end  post  may  be  found  by 

SG  =  ~   •   Ss     ........    (22) 

In  Fig.  8  is  shown  the  influence  line  for  the  stress  $4 
in  any  vertical  post.  The  influence  ordinates  are  deter- 
mined by  taking  moments  about  the  intersection  of  the 
upper  and  lower  chord  members  which  are  cut  by  the 
section.  The  algebraic  values  of  these  ordinates  are  trans- 
formed by  use  of  easily  discovered  relations  between  sim- 
ilar triangles.  The  slopes  and  coefficients  are  then  calcu- 
lated in  the  usual  way.  The  influence  line  indicates  that 
the  live  load  should  advance  into  but  not  beyond  the  panel 
p  for  a  maximum  compression,  and  for  this  reason  MI  and 
If  2  equal  zero  for  the  usual  case.  The  numerical  value  of 


34  LIVE-LOAD    STRESSES 

the  maximum  compression  S*  in  a  vertical  post  is,  therefore, 


The  coefficients  for  the  stress  in  any  inclined  web  mem- 
ber are  given  by  Fig.  9.     The  quantities  for  Si  and  >S2  are 


Ordinates 


Slopes 


Coefficients 


.  _ 

I  b.L bL 


A. 

bl 


bLp\ 


a 
bL 


d 
bl 


. 

(>P\ 


_ 

IP 


FIG.  8. 

as  shown,  and  the  quantities  for  S3  are  of  the  same  alge- 
braic form  except  that  they  are  of  opposite  sign  through- 
out. For  the  usual  position  of  the  live  load  advanced  from 
the  right  into  but  not  beyond  the  panel  p  for  maximum 
stress,  the  moment  sums  Mi  and  M2  equal  zero,  and  the 
numerical  values  of  the  maximum  tension  Si  and  &>  and 
of  the  maximum  compression  &  are  given  by  the  following 
formula: 


(24) 


LIVE-LOAD    STRESSES 


35 


In  a  special  case  where  the  loading  must  be  advanced 
beyond  the  panel  p  until  the  tension  in  the  inclined  counter- 
web  member  $2  is  balanced  by  the  dead-load  compression 


FIG.  9. 

in  this  same  member,  the  value  of  M2  is  not  zero,  and  the 
formula  for  $2  becomes 


Or,  letting  Mc  = 


-  S). 


36 


LIVE-LOAD    STRESSES 


Note  that  the  coefficients  of  M4  and  Mc  in  this  formula 
are  the  same  as  the  coefficients  for  M4  and  M3  in  formula  (24) . 

The  influence  line  for  the  counter-tension  in  a  vertical 
post  is  shown  in  Fig.  10.  For  the  usual  case,  the  loading 
advances  beyond  the  panel  but  not  beyond  the  end  of  the 
span.  Therefore  Mi  is  equal  to  zero,  so  that 


FICJ.  10. 


(26) 


where  K  and  M0  stand  for  the  corresponding  terms  in  the 
parentheses.  In  order  that  T  be  a  maximum  the  live  load 
must  advance  beyond  the  position  for  the  maximum  tension 
&  until  the  tension  as  computed  by  formula  (25)  becomes 
equal  to  the  dead-load  compression  in  this  same  member. 
For  this  position  of  the  live  load,  the  value  of  T  is  then 
computed  by  using  formula  (26).  It  may  be  noted  that 


LIVE-LOAD    STRESSES  37 

some  specifications  state  that  only  %  of  the  dead-load 
compression  is  to  be  counted  as  effective  in  counteracting 
the  live-load  tension  in  an  inclined  counter-web  member. 
This  specification  has  been  observed  in  the  problem  to 
follow. 

A  review  of  the  preceding  formulas  shows  that  all  the 
live-load  stresses  may  be  computed  by  formulas  (21),  (22), 
(23),  and  (24),  except  the  counter-tension  in  a  vertical  post 
and  the  tension  in  a  floor-beam  hanger.  Formula  (25)  makes 
it  possible  to  find  readily  by  trial  the  position  of  loading  for 
maximum  counter-tension  in  a  vertical  post,  and  formula 
(26)  gives  the  value  of  this  tension.  The  maximum  ten- 
sion in  the  floor-beam  hanger  may  be  found  by  the  use  of 
formulas  (14a)  and  (15a)  for  pier  reaction  between  equal 
spans. 

If  the  chords  of  the  Pratt  truss  are  parallel,  there  will 
be  no  counter-tension  in  any  vertical  post.  Formula  (21) 
for  the  stress  in  a  horizontal  chord  member  and  formula 
(22)  for  the  stress  in  the  inclined  end  post  remain  unchanged. 
Formulas  (23)  and  (24)  for  web  stresses  are  simplified  be- 
cause a  =  b  =  depth  of  truss. 

The  formulas,  therefore,  for  the  Pratt  truss»with  parallel 
chords  are: 
Stress  in  horizontal  chord  members  = 


Stress  in  inclined  end  post  =  S&  =  -  $5  .......   (22) 


Stress  in  vertical  post  =  £4  =     j$£*  "    ~M^  -    -    -    (29) 
Stress  in  inclined  web  member  = 

)*'-;&  ••  •  •  (30) 

One  general  formula  will  suffice  for  finding  the  position 
of  loading  for  maximum  chord  and  web  stresses  of  a  Pratt 
truss  with  either  inclined  or  parallel  chords.  The  formulas 


38 


LIVE-LOAD    STRESSES 


(21),  (23),  (24),  (29),  and  (30)  for  these  stresses  are  of  one 
general  form 

S  =  (G)  M,  -  (H)  M,      (27) 

where  G  and  H  are  the  corresponding  coefficients  of  M4 


FIG.  11. 

and  Ms  in  the  preceding  formulas.     The  rate  of  variation 
of  S  as  the  load  advances  is 

^  =  GWt  -  HW3  =  H  (|  Wt  -  W3)    .    .   (28) 

When  any  one  of  the  above  stresses  is  a  maximum,  the 

CC1  \ 

jjWi  —  W3J  passes  through  zero  as  a  wheel  is 

shifted  from  right  to  left  of  the  salient  point  3  in  Figs. 
7,  8,  or  9. 

The  preceding  formulas  for  the  live-load  stresses  are 
summarized  for  convenient  reference  in  Art.  11  preceding 


LIVE-LOAD    STRESSES  39 

the  Tables.  The  important  dimensions  and  quantities  in 
Figs.  7,  8,  and  9  are  summarized  in  Fig.  11.  If  a  uniform 
live  load  is  used,  the  shaded  areas  in  Fig.  lla,  b  and  c  mul- 
tiplied by  the  intensity  of  the  uniform  load  will  give  the 
maximum  live-load  stresses.  The  algebraic  value  of  any  one 
of  these  triangular  areas  is  conveniently  expressed  as  the 
base  of  the  triangle  times  y%  of  the  given  algebraic  ordi- 
nate.  The  lengths  of  the  bases  of  the  shaded  areas  in  Figs, 
lla  and  b  may  be  readily  determined  by  one  of  the  con- 
structions shown  in  Figs.  12a  and  12b,  which  give  the  po- 
sition of  the  unit  load  for  zero  stress  in  the  members  indi- 
cated. The  proofs  that  these  constructions  locate  neutral 
points  are  not  given,  for  they  are  generally  known,  and 
are  proved  in  numerous  texts  on  bridges.  (See  Marburg's 
"  Framed  Structures  and  Girders,"  Vol.  I,  page  392.) 

The  application  of  the  preceding  formulas  will  now  be 
made  to  the  calculation  of  the  live-load  stresses  in  the  two 
single  track  through  Pratt  trusses  shown  in  Figs.  13  and 
14.  A  convenient  procedure  is  as  follows: 

1.  Determine  the  lengths  of  all  inclined  members  and 
write  then-  values  on  the  truss  outline. 

2.  Determine  the  values  of  the  intercepts  a  as  defined 
by  Fig.  11  and  write  their  values  on  the  truss  outline. 

3.  Write  on  the  truss  outline  the  distances  of  the  sev- 
eral panel  points  from  the  right  end  of  the  span. 

4.  Write  down  the  reciprocals  of  the  span,  panel  length, 
and  lengths  of  vertical  members. 

5.  Make  a  form  for  tabulating  calculations  and  list 
members  in  some  convenient  form  as  is  done  in  Figs.  13 
and  14. 

6.  Calculate  the  numerical  values  of  the  coefficients  G 
and  H  for  the  several  members  by  use  of  the  formulas 
already  derived. 

7.  Determine  the  position  of  the  loading  for  maximum 

/  C1  \ 

stress  by  finding  the  position  of  loading  causing  (-77  TF4  —  TF3  J 


40 


LIVE-LOAD    STRESSES 


to  pass  through  zero,  and  for  this  position  of  loading  select 
from  Table  2  the  corresponding  values  of  M4  and  M3.    At 


VARIOUS  CONSTRUCTIONS  USED  TO  FIND    NEUTRAL  POINTS  IN  PRATT  TRUSSES. 


U,  U2  U,,  U4  U, 


the  same  time  tabulate  the  length  Z/i  of  loading  causing 
maximum  stress  as  this  value  is  used  in  the  impact  formula 


LIVE-LOAD    STRESSES 


41 


/  =  S' 


300 


Li  +  300* 


8.  Calculate  values  of  S  =  GM4  —  HMS  and  combine 
with  impact  and  dead-load  stresses.  When  the  dead-  and 
live-load  stresses  are  of  opposite  sign,  the  combination  is 
usually  not  algebraic  but  according  to  the  particular  speci- 
fication that  is  used. 


a  =  38' 
2G.OO 


26.08 


A          1  2  a 

1-4- 208  =  .00480 
1-^  26  =.0385 
Span  208'  Live  Ld.  E  J3 


-f-  32  =.03125 
-^36  —.02778 


FIG.  13. 


6  7  A 

l-4-38=.0£C32 


>/r*%wv» 

-rcru^^i 

M 

•KJT 

OMj 

HMs 

T  . 

300 

j 

DL 

Total 

Mem. 

w  neci 

4 

HM 

\jrIYl4 

Xijyi3 

Mm 

Li+300 

K 

EF 

00373 

.0385 

3@3 

33970 

287 

127 

11 

-116 

143 

.677 

-  78 

-  40 

-234 

ED 

00481 

.0442 

3@2 

46255 

287 

223 

13 

+210 

169 

.640 

+134 

+  83 

+427 

GH 

00405 

.0385 

2@4 

21531 

100 

87 

4 

-  83 

112 

.728 

-  60 

-  15 

-158 

GF 

00500 

.0450 

3  @3 

33970 

287 

170 

13 

+157 

143 

.677 

+106 

+  48 

+311 

IJ 

00480 

.0385 

2  @5 

12940 

100 

62 

4 

-  58 

86 

.777 

-  45 

+    7 

IH 

00580 

.0466 

3  @4 

23375 

287 

136 

13 

+123 

117 

.719 

+  88 

+  21 

+232 

JK 

00580 

.0466 

2  @5 

12940 

100 

75 

5 

+  70 

86 

.777 

+  54 

-  21 



ML 

00777 

.0493 

2  @6 

6550 

100 

51 

5 

+  46 

60 

.833 

+  38 

-  50 

NO 

01030 

.0496 

2  @7 

2307 

100 

24 

5 

-  19 

34 

.898 

-  17 

+  83 

No 

counter 

AC  =AD 

00390 

.0312 

4@1 

63111 

600 

247 

19 

+228 

200 

.600 

+137 

+101 

+466 

BC 

-362 

-217 

-160 

-739 

AF 

00695 

.0278 

7@2 

59095 

2694 

410 

75 

+335 

193 

.608 

+203 

+154 

+692 

BE 

-339 

-206 

-156 

-701 

AH 

00985 

'0263 

li@3 

59661 

7310 

587 

192 

+395 

194 

'eo7 

+239 

+181 

+815 

BG 

-396 

+240 

-181 

-817 

BI 

01315 

'0263 

13®  4 

50901 

9585 

670 

252 

-418 

178 

^627 

-262 

-194 

-874 

CD 

0385 

.0770 

4  @  1 

3725 

600 

144 

46 

+  98 

44 

.872 

+  86 

+  25 

+209 

Post 

at 

Mem. 

M4 

Me 

S 

IB 

K 

M,,      T 

Li 

300 

I 

D.L. 

Total 

Li+300 

5 

JK 

22261 

2390 

+  16 

__  

-14.0020. 

311340   +23 

114 

.725 

+17 

+3 

+  43 

6 

ML 

8865 

687 

+35|-34i  0021 

1  5960   +13 

71 

.8 

+10 

+1 

+  24 

42  LIVE-LOAD    STRESSES 

9.  Find  positions  of  loading  for  maximum  counter-ten- 
sions in  posts  and  compute  values  by  use  of  formulas  (25) 
and  (26). 

PROBLEM  1. 

Calculation  of  Live-load  Stresses  in  a  Pratt  Truss  with 
Inclined  Chord. 

The  complete  data  for  this  problem  are  given  in  Fig. 
13.  Items  1  to  5  of  the  above  method  of  procedure  need 
no  explanation.  The  values  of  the  coefficients  G  and  //, 
the  position  of  the  loading  for  maximum  stress,  and  the 
value  of  the  maximum  stress  will  be  determined  for  several 
typical  members;  for  example,  vertical  post,  inclined  web 
members,  horizontal  chords,  end  post,  and  inclined  chords. 

Vertical  Post  EF. 
Formula  S,  =  (^r)  M4  -  (-)  M3  .    .    .    .    .    .   (23) 

Refer  to  Fig.  11  for  definition  of  dimensions. 
G  =  £-  =  |f  (.00480)  =  .00373 

DLi         oO 

H  =  -  =  .0385 
Try  ws  at  panel  point  3.     Use  Table  2.     L±  =  143'. 


Therefore  w3  at  3  gives  a  maximum. 

S  =  GM,  -  HM3  =  .00373(33970)  -  .0385(287.5) 
=  126.7  --  11.0  =  115.7" 

-  300  300        A77 

Impact  factor  ==  --       =-          ==  .677 


Impact  stress  =  .677  X  115.7  =  78.3fc. 


LIVE-LOAD    STRESSES  43 

Inclined  Web  Member  ED. 
Formula  8l  =  (^)  M4  -  (£)M,     ....   (24) 

Refer  to  Fig.  11  for  definition  of  dimensions. 


H  =  ~  =  —~  (.0385)  =  .0442 

Try  ws  at  panel  point  2.     Use  Table  2.     Za  =  169'. 

nnj.si  37.5       -f- 

=  ^^T  (505.0)-    or    =or 

.U44^  £}o  c 

D^.O          — 

Therefore  w3  at  2  gives  a- maximum. 
iS  =  GM*  -  HMS  =  .00481(46255)  -  .0442(287.5) 
-  223  -  13  =  210fc. 

300 
Impact  factor  =  -j™  =  .640 


Impact  stress  =  .640  X  210  =  134*. 

Inclined  Web  Member  ML. 
Formula  S,  =  4  -  jtf,     ...      .(24) 


Refer  to  Fig.  9  or  Fig.  11  for  definition  of  dimensions. 


/         4fi  04 

H  =  ^  =  -^-  (-0385)  -  -0493 

Try  w2  at  panel  point  6.     Use  Table  2.     la  =  60'. 

00777  1^.5     + 

w(19°)-3-  =  °_r 

Therefore  w2  at  6  gives  a  maximum. 


44  LIVE-LOAD    STRESSES 


S  =  GM,  -  HM,  =  .00777(6550)  -  .0493(100) 


=  51  --  5  =  46*. 

300 
Impact  factor  =          =  .833 


Impact  stress  =  .833  X  46  =  38*. 
Lower  Chord  Member  AC  —  AD. 

Formula  &  ==  Q^4  -  (±)jf,   .....    (21) 

Refer  to  Fig.  11  for  definition  of  dimensions. 

syyi 

G  =  ~  =  J  (.03125)  =  .00390 

H=  —  =  .0312 

v 

Try  w*  at  panel  point  1.  .  Use  Table  2.     L,  =  200'. 


Therefore  io4  at  1  gives  a  maximum. 
S  =  GM4  -  #Af,  =  .00390(63111)  -  .0312(600) 
=  247  --  19  =  228*. 

Impact  factor  =          =  .600 


Impact  stress  =  .600  X  228  =  137*. 

End  of  Post  BC. 
Formula  S6  =  -^  S-0    .........    (22) 

SG  =  ^l3  (228)  =  362*,  and  impact  =  ^p  (137)  =  217*. 

Lower  Chord  Member  AH. 
Formula  S5  =  M4  -         M3    .    .    .    .   (21) 


LIVE-LOAD    STRESSES  45 

Refer  to  Fig.  11  for  definition  of  dimensions. 

/yyi 

G  =  ~  =  |  (.02632)  =  .00985 

H  =  -i-  =  .0263 

Try  wu  at  panel  point  3.     Use  Table  2.     Li  =  194'. 
ri  190 

l^- 


Therefore  MU  at  3  gives  a  maximum. 
S  =  GM4  -  HM3  =  .00985(59661)  -  .0263(7310) 
=  587  --  192  =  395*. 

Impact  stress  =         S  =  .607  X  395  =  239*. 


Top  Chord  Member  BG. 
Formula  SG  =  —  S,     ..........    (22) 

S.  =  ^  (395)  «.  396fc. 

2fi  AQ 

Impact  =    ^  (239)  =  240". 

Counter-Tension  in  Post  at  Panel  Point  5. 
Formulas 
S2  =  Stress  JK  = 


=  tension  in  post. 

-)(-n^-^)=K-M0     (26) 

Refer  to  Fig.  10  for  definition  of  dimensions. 

The  calculation  of  the  dead-load  compression  in  JK  is 


46  LIVE-LOAD    STRESSES 

not  given,  but  the  value  is  21fc.  Two-thirds  of  this  com- 
pression, or  14fc,  will  be  considered  effective  in  counterbal- 
ancing the  live-load  tension  in  JK.  The  live  load  must  be 
advanced  beyond  the  position  of  maximum  live-load  ten- 
sion in  JK  (i.e.,  w2  at  panel  point  5)  until  &,  or  the  stress 
in  JK,  equals  14fe.  This  must  be  done  by  trial,  Sz  being 
figured  each  time  by  formula  (25).  It  is  found  that  when 
114'  of  loading  has  advanced  upon  the  bridge,  this  condi- 
tion is  approximately  satisfied.  For  this  position  of  loading 

M*  =  22261 

Mc  =     MS-    M2    =  (2565  -  175)  =  2390 

-«»» 


Therefore, 

&  =  .00580(22261)  -  .0466(2390)  =  16*. 
This  value  of  S*  =  16fc  balances  %  D  =  -  14fc,  nearly 
enough  for  practical  purposes.     Therefore,  compute  T  for 
this  position  of  the  live  load. 


K  =        =  -00203 


M0  =  $/8  (22261)  -  2565  =  11340 
T  =  .00203(11340)  =  23* 

300 
Impact  factor  =    T    =  .725 


Impact  stress  for  T  =  .725  X  23  =  17fc. 

PROBLEM  2. 
Live-load  Stresses  in  a  Pratt  Truss  with  Parallel  Chords. 

The  complete  data  for  this  problem  are  given  in  Fig. 
14.     Formulas  (21),  (29),  and  (30)  give  the  values  of  the 


LIVE-LOAD    STRESSES 


47 


coefficients  G  and  H,  which  are  identical  for  several  mem- 
bers of  any  Pratt  truss  with  parallel  chords.  The  proce- 
dure for  finding  the  positions  of  the  loading  and  maximum 
stresses  is  exactly  as  in  Problem  1.  It  should  be  noted 
that 


Stress  FG  =  Stress  EF  X 


37.54 


HI  = 


BC  = 


(C 


GHX 


ACX 


25 


Live  Load  E  50 


3 

A 
•£-=•^=.00667 

^  =~j|f=='0400 

Fig.  14. 


Secant « 


Mem. 

G 

H 

Wheel 

M4 

Ma 

S 

CD 
EF 
FG 

.0400 
.00667 

.0800 
.0400 

4@1 

3  "  3 

3564 
13520 

600 

287 

95 
79 
106 

GH 
HI 

.00667 

.0400 

2  "  4 

6170 

100 

37 
50 

JK 
DE 
BC 

.00894 
.  00894 

.0536 
.0536 

2  '  5 
3  '  2 

2179 
21895 

100 

287 

14 

181 

272 

AC  =  AD 
AF  =  BE 
BG 

.  00595 
.01190 
.  01785 

.0357 
.0357 
.0357 

4  '  1 
7  '  2 
12  '  3 

33970 
31375 
34411 

600 
2694 

8385 

181 
278 
314 

The  stresses  in  all  of  the  chord  members  may  be  checked 
by  use  of  Table  8,  and  the  stresses  in  the  end  post  and  web 
members  may  be  checked  by  Table  9.  The  stress  in  CD 
agrees  with  the  maximum  pier  reaction  in  Table  7.  Table 
3  may  be  used  to  find  the  position  of  loading  for  maximum 
chord  stresses,  and  Table  6  gives  position  of  loading  for 
maximum  web  stresses. 


ARTICLE  VIII. 

THREE-HINGED  ARCH.    APPLICATION  OF  THE  GENERAL  METHOD 
TO  THE   CALCULATION  OF  LIVE-LOAD   STRESSES. 

THE  general  formulas  -r-  =  2CW  and  S  =  2 CM  may 
be  used  directly  to  find  the  position  of  loading  and  the 


FIG  15. 

value  of  the  maximum  live-load  stress  in  any  member  of 
a  framed  structure  as  soon  as  the  influence  line  for  this 
member  and  the  ordinates  at  all  salient  points  have  been 

48 


LIVE-LOAD    STRESSES 


49 


determined.  This  method  is  applied  to  the  calculation  of 
maximum  live-load  stresses  for  the  three-hinged  arch  shown 
in  Fig.  15.  Cooper's  E40  loading  is  used. 

First  are  drawn  the  influence  lines  for  the  horizontal 
and  vertical  components  of  the  reaction  at  the  left  hinge. 
The  vertical  component  Vi  is  the  same  as  for  a  simple 
span  L.  The  horizontal  component  Hi  equals  the  bending 
moment  at  the  centre  of  the  span  L  divided  by  the  depth  h. 
The  influence-line  ordinates  for  all  members  are  now  found  by 
drawing  five  Maxwell  diagrams,  one  of  which  is  reproduced 
in  Fig.  16.  From  the  influence  lines  for  Vi  and  Hi,  the 
value  of  Vi  is  .8889  and  Hi  is  .2187  for  a  one-pound  load 
at  Ui.  The  external  loads  acting  on  the  left  half  of  the 
arch  are  then  as  shown  in  Fig.  16a.  The  load  line  axbcya 
in  Fig.  16b  is  drawn  to  a  scale  of  10"  =  1  pound,  and  the 
Maxwell  diagram  completed  in  the  usual  way.  The  scaled 

TABLE  A 
INFLUENCE-LINE  ORDINATES  FOR  THREE-HINGED  ARCH 


Members 

ORDINATES 

1  Ib.  at  Ui 

1  lb.  at  U2 

l.lb.  at  Us 

1  lb.  at  U4 

1  lb.  at  U'4 

E/ot/i=  
UiUt=  
U*U3  =  
U3U*  =  

-  .403 
-  .417 
-  .378 
—  .171 

-  .223 
-  .833 
-  .756 
-  342 

-  .045 
-  .286 
-1.135 
—  513 

4-  .130 
+  .262 
+  .189 
—  685 

+  .201 

+  .477 
4-  .757 
-j-  548 

LoLi  =  

—  .295 

-  590 

-  885 

-1  180 

—  1  182 

LxL2=  
LzL3-  . 

+  .221 

4-  217 

-  .264 

4-  434 

-  .740 
—  408 

-1.224 
—  1  248 

-1.302 
1  484 

L3L4-  . 

4-  164 

4-  328 

4-  491 

—  1  086 

1  674 

L4L5  -  . 

—  048 

—  096 

—  145 

—  193 

—  1  420 

t/oLo  -  . 

—  €92 

—  384 

—  075 

4-  234 

_|_  345 

C/iLi  =  . 

—  1  014 

—  632 

—  253 

4-  129 

4-  287 

U'l  — 

4-  022 

—  955 

—  490 

—  043 

+  165 

U3L3  =  . 

4-  075 

4-  150 

—  775 

—  317 

—  076 

C/4^4  =  .  . 

+  114 

4-  226 

4-  342 

—  545 

—  364 

C/oI/i  =  . 

+  800 

4-  441 

4-  085 

—  270 

—  400 

UiL»-.  . 

4-  019 

4-  878 

4-  350 

—  180 

—  398 

U2L3  -  .  .   . 

-  044 

—  088 

4-  986 

"4.  086 

—  324 

U3L4  =  

—  221 

—  442 

—  662 

+  928 

4-  224 

f/4L5=  

H  .. 

-  .206 
0  2187 

-  .412 
0  4375 

-  .617 
0  6562 

-  .823 
0  8750 

+  .657 
0  87^0 

V  .. 

0  8889 

0  7777 

0  6666 

0  5555 

0  4444 

6 

14° 

29° 

44° 

58° 

63° 

50 


LIVE-LOAD    STRESSES 


values  of  these  stresses  are  the  influence  ordinates  for  a 
one  pound  load  at  U\.  In  an  exactly  similar  way  the  in- 
fluence ordinates  for  a  unit  load  at  U2,  Us,  U4,  and  U\  are 
determined.  The  influence  lines  are  straight  from  U'Q  to 


(a) 


FIG  16. 

£7'4.  Table  A  gives  the  influence  ordinates  for  all  members 
and  also  for  the  horizontal  and  vertical  components  of  the 
reaction  at  the  left  hinge.  The  angle  6  is  the  inclination 
of  this  reaction  with  the  vertical. 

The  calculation  of  the  live-load  stresses  in  any  one  mem- 
ber is  typical.  The  member  UJL±  is  taken.  The  influence 
line  for  this  member  is  drawn  to  scale  in  Fig.  15  by  use  of 
the  influence  ordinates  from  Table  A.  The  salient  points 
occur  below  panel  points  C73,  Ut,  and  U\.  The  distance 


LIVE-LOAD    STRESSES  51 


662 
from  U3  to  the  neutral  point  0  equals  AAO  4.  oog 


Calculation  of  Slopes. 
Slope  of  df  =  0 


jy  - 

hk  = 

km  = 
mn  = 

68 
-  .662  -  (.928) 

T  .\JL\JU 

-  .0758 
+  .0336 

21 
.928  -  (.224) 

21 
.224  -  0 

84 
0 

Calculation  of  Coefficients. 

Cl  =  0  -  (.0105)               -  .0105 

C2  =  .0105  -  (  --  .0758)  -  +  .0863 

C8  =  -  .0758  -  (.0336)               -  .1094 

C4  =  .0336  -  (.0027)        =  +  .0309 

C5  =  .0027  -          0          =  +  .0027 

The  sum  of  these  coefficients  equals  zero.     This  agrees 
with  formula  (6)  of  Art.  3. 

It  should  be  remembered,  as  is  pointed  out  in  Art.  3, 
that  the  value  of  these  coefficients  may  be  measured  graph- 

2  59 
ically.    For  example,  in  Fig.  15  the  value  of  C2  is  -r-  = 


.0863. 


7O 

By  use  of  the  formula  -,-  =  SCTF  and  Rule  1  of  Art. 


3,  the  position  of  loading  for  maximum  tension  in 
may  now  be  determined.  Try  wheel  3  at  t/4  with  the  load- 
ing advancing  toward  the  left.  Take  the  values  of  the  load 
sums  and  moment  sums  for  #40  from  Table  2. 


52  LIVE-LOAD    STRESSES 

|~  =  2CW  =  -  .1094(30)  +.309(103)  +.0027(302)  =  +  .7 
-.1094(50)  +  .309(103)  +.0027(302)  =  -  .7 


Therefore  w3  at  17*  gives  a  maximum  tension  in 
and  its  value  is 

S  =  zCM  =  -.1094(230)  +  .309(1846)+.0027(19001)=83fc. 

TOf 

By  use  of  the  formula  -      =  ?CW  and  Rule  2  of  Art.  3, 


the  position  of  loading  for  maximum  compression  in 
is  now  determined.  Try  wheel  2  at  Us  with  the  loading 
advancing  toward  the  right.  Note  that  the  signs  of  the 
coefficients  remain  unchanged.  Take  the  values  of  the  load 
sums  and  moment  sums  for  #40  from  Table  2. 


-.0105(192)  +  .0863(10)  =  --  1.3 
-       =  SCTF  =  -.0105(192)  +  .0863(30)  =  +  0.6 


Therefore  w2  at  Us  gives  a  maximum  negative  stress,  or 
compression,  in  USL^  and  its  value  is 

S  =  ?CM  =  -.0105(7092)  +  .0863(80)  =  -  67*. 

The  above  values  of  83fc  and  67fc  for  maximum  tension 
and  compression  in  C73L4  may  be  checked  by  use  of  formula 
S  =  qAz  (2),  the  values  of  q  being  taken  from  Table  16. 

Tension  U&L*  by  Equivalent  Uniform  Load. 
The  area  of  the  tension  part  of  the  influence  line  equals 
Az=  27.2 

The  influence  line  ohkm  is  not  triangular,  but  a  tri- 
angular influence  line  with  intervals  Zi  =  10  ft.  and  12  = 
45  ft.  approximates  its  shape  closely  enough  for  the  selec- 
tion of  an  equivalent  uniform  load.  For  Zi  =  10r  and  Z2  = 
45',  Table  16  gives  3.080fc  as  the  equivalent  uniform  load. 


LIVE-LOAD    STRESSES  53 

Therefore, 


S  =  qAz  =  (3.080)  (27.2)  =  84. 

This  value  checks  very  closely  that  obtained  by  the 
exact  method. 


Compression  U^L^  by  Equivalent  Uniform  Load. 

Choose  from  Table  16  the  equivalent  uniform  load  for 
h  =  10  ft.  and  Z2  =  65  ft.  From  the  influence  line  Az  = 
23.7. 

Therefore, 

S  =  qAz  =  (2.870)  (23.7)  =  68fc. 

This  checks  closely  the  value  obtained  by  the  exact 
method. 

Calculation  of  other  members  of  this  arch  and  of  some 
more  complicated  framed  structures  shows  a  close  agree- 
ment between  the  two  preceding  methods.  The  latter 
method  is  the  simpler  when  a  table  of  equivalent  uniform 
loads  has  been  made,  especially  in  the  case  of  the  more 
complex  influence  lines  for  members  of  swing  bridges,  two- 
hinged  arches,  arch  ribs,  etc.  The  method  of  calculating 
a  table  of  equivalent  uniform  loads  will  be  explained  in  the 
following  article. 


ARTICLE   IX. 

EQUIVALENT   UNIFORM   LOADS. 

AN  equivalent  uniform  load  is  one  which  gives  the  same 
stress  as  does  a  loading  which  is  not  uniform.  For  any 
given  standard  loading,  the  equivalent  uniform  load  is  dif- 
ferent for  stresses  whose  influence  lines  differ.  Since  the 
forms  of  influence  lines  are  innumerable,  a  table  of  exact 
equivalent  uniform  loads  for  all  stresses  is  impracticable. 
A  table  of  equivalent  uniform  loads,  however,  for  stresses 
whose  influence  lines  are  triangular  may  be  used  with  little 
error  in  selecting  equivalent  uniform  loads  for  stresses  whose 
influence  lines  are  not  triangular.  It  is,  therefore,  sufficient 
for  practical  purposes  to  make  tables  of  equivalent  uniform 
loads  for  a  series  of  triangular  influence  lines.  It  may  be 
shown  that  the  equivalent  uniform  load  for  any  triangular 
influence  line  is  dependent  entirely  upon  the  intervals  Zi 
and  12)  and  is  independent  of  the  ordinate  h  at  the  apex 
of  the  influence  line.  Consider  the  triangular  influence  line 
in  Fig.  Ib  to  be  for  any  stress  S.  Let  the  ordinate  below 
C  be  any  value  h.  If  q  equals  the  equivalent  uniform  load 
covering  li  and  lz, 

S 
S  =  qAz,  or  q  =  j-       (A) 

The  area  of  this  influence  line  is 

Az  =  \  (h  +  «  =  \L (B) 

Furthermore,  if  the  concentrated  live  loads  have  been 
placed  so  as  to  give  the  maximum  pier  reaction  between 
two  spans  h  and  k,  this  same  position  of  loading  will  give 
maximum  S,  if  the  influence  line  for  S  is  a  triangle  with  the 

54 


LIVE-LOAD    STRESSES  55 

same  intervals  li  and  Z2.  Since  the  influence  ordinates  for 
S  are  related  to  the  influence  ordinates  for  R  as  h  is  to 
unity, 

S         h 
R  ==  1.00 

Or 

S  =  hR      .....    .    .    .    .    (O 

Substituting  the  values  of  Az  and  S  from  equations  (B) 
and  (C)  in  equation  (A), 

q  =  hR  +     L~      ......   (D) 


It  appears,  therefore,  that  q  is  independent  of  /t. 
From  formula  (16)  of  Art.  5, 


Li  t-2 

Substituting  for  R  in  equation  (D), 

2/2       2M 


(16) 


(3D 


The  term  M  is  the  bending  moment  in  the  span  L  = 
li  +  12  at  the  point  where  the  intervals  are  h  and  Z2. 

Tables  (10)  to  (18)  inclusive  have  been  calculated  for 
the  positions  of  the  live  load  given  by  Table  3.  The  values 
of  M  were  first  found,  then  the  values  of  R,  and  finally  the 
values  of  the  equivalent  uniform  loads.  The  three  formulas 
that  were  used  in  succession  are 

If  -  jJMt+jJfi-  Jfi     .....   (10) 
R  =  /y  M       ..........   (16) 

tl  t-2 

2M      2R 


56  LIVE-LOAD    STRESSES 

An  example  of  the  use  of  equivalent  uniform  loads  has 
already  been  given  in  Art.  8.  The  general  formula  S  = 
qAz  may  be  used  in  any  case.  For  the  special  cases  of 
bending  moment  in  a  beam  and  pier  reaction  between  two 
simple  spans,  formula  (31)  gives 


<32> 


The  quantities  in  the  parentheses  are  the  areas  of  the 
influence  lines  for  M  and  R  respectively. 


ARTICLE  X. 

METHOD    OF    CALCULATING   TABLE    OF    LOAD    SUMS    FOR    ANY 
STANDARD    LOADING.       ILLUSTRATIVE    EXAMPLE. 

THE  definitions  of  moment  sum  and  load  sum  are  given 
at  the  beginning  of  Art.  2.  It  is  at  once  evident  that  a 
table  of  load  sums  may  be  computed  by  adding  the  succes- 
sive loads.  It  may  be  shown  that  the  table  of  moment 
sums  may  also  be  calculated  by  the  process  of 
addition. 

From  formula  (5a)  of  Art.  2, 


CnWn     = 


dMa 


dx 

Or 

dMa  =  Wa  -  dx. 

Expressed  in  words,  the  increase  in  the  moment  sum  for 
an  increase  dx  in  the  distance  of  the  centre  of  moments 
from  wheel  1  equals  the  load  sum  times  dx.  If  the  load 
sum  is  constant  for  an  interval  dx  =  1  foot,  as  between  con- 
centrated loads,  the  increase  of  the  moment  sum  for  dx  = 
1  foot  equals  the  corresponding  load  sum.  If  the  load  sum 
is  not  constant,  but  uniformly  increasing,  as  when  the  cen- 
tre of  moments  lies  within  the  uniform  load,  the  increase 
of  the  moment  sum  for  dx  =  1  foot  equals  the  average  value 
of  the  load  sum  for  this  one  foot  interval.  The  appli- 
cation of  the  foregoing  principles  is  made  clear  by  the  fol- 
lowing example. 

Example. — Give  explicit  directions  for  the  calculation  of 
a  table  of  load  sums  and  moment  sums  at  intervals  of  1 
foot  from  IT  to  400'  for  Cooper's  #40  loading. 

Solution. — Calculate  the  table  of  load  sums  by  adding 

57 


58  LIVE-LOAD    STRESSES 

the  loads  one  by  one,  taking  a  sub-total  for  each  addition. 
Thus,  the  following  numbers  are  added,: 

1—10 

4— 20's 
4—  13's 
1—10 
4— 20's 
4—  13's 
391—  2's 

If  the  final  total  checks  284  +  391  X  2  =  866,  the  table 
of  load  sums  is  correct. 

Assume  now  that  the  table  of  load  sums  for  E40  has 
been  completed.  The  table  of  moment  sums  may  now  be 
found  as  directed  below.  The  following  numbers  are  to 
be  added  one  by  one,  taking  a  sub-total  for  each  addition: 

8— 10's 
5— 30's 
5— 50's 
5— 70's 
9— 90's 

5—  103's 

6—  116's 
5—  129's 
8— 142's 
8— 152's 
5— 172's 
5— 192's 
5— 212's 
9— 232's 
5— 245's 
6— 258's 
5— 271's 
5— 284's 
1—285 
1—287 
1—289 


and  all  odd  numbers  up  to  865. 

If  the  final  total  checks  up  183,689,  which  is  figured 
independently,  the  table  of  moment  sums  is  correct. 

The  preceding  additions  may  be  made  most  satisfac- 
torily on  a  recording  adding  machine.  Table  2  was  cal- 
culated in  this  way. 

It  will  be  noted  that  the  table  of  load  sums  serves  as  a 
table  of  differences  for  the  table  of  moment  sums. 


ARTICLE  XI. 

SUMMARY   OF   FORMULAS. 

Art.  1. 

z  =  zwz    .  .  '.  .  .  .;;•  ...  ..  v  ..-.-  .  .  (i) 


Z  =  qAz       .....  .    .    .,.    .    .  (2) 

.  Z  =  w2z      .......    .......  (3) 

Z  =  z2w  =  zW  .    .    .  V  .    .    ......  (4) 

Art.  2. 

Z  =  2wnzn  =  Ca2wnxn  =  CaMa    .....  (5) 
dZ                     d(CaMa)       CadMa 


dx         aa  dx  dx 


Art.  4.     Girder  Bridge  without  Panels. 
End  reactions. 


Art.  3. 

ZC  =  0      .....  ../;    .    .,  ,  ;:!    ....   (6) 

S  =  2CM    .    .   /.    .    .    .   .......   (7) 


I  ......    . 

R^=Ws_^-M,      .•;•....'.;.   (9a) 
Bending  moment  for  unequal  segments  li  and  1%. 

M  =     Klfa  +  --  Afi  -  M2     ,  ;   .    .    .  .  ,    (10) 


Wt-W,      .    .....  (11) 

59 


60  LIVE-LOAD    STRESSES 

Bending  moment  at  centre,     h  =  Z2  =  ~^~ 

-.      M3  +  Mi 

M  =  o ^/2     •  -•    • 


dx  2 

Shear  at  any  section. 

s-*& 


Location  of  centre  of  gravity  of  loading  on  span. 
M3  -  Mi  -  LWl 


When  Mi  =  0, 

5  =  |£     . (13a) 

Ar£.  5.     Pier  Reaction. 
For  unequal  spans  Zi  and  Z2. 


cffi     .  Ej  .   Ej      Aw  _  J 
aic         Z2          'i        ZiZ2  Z: 

For  equal  spans  Zi  and  k  equal  to  Z. 

R  =      '^     g1"      -2 (14a) 

m  =  TF3  +  ^-2^     -    -    :    -    -   (16a) 
Relation  between  R  and  M , 

R  =  rrM  .  (16) 


Art.  6.     Girder  Bridge  with  Panels. 
Shear  in  end  panel;   general  case. 

=  ^Mz+^M^Mz  — 


LIVE-LOAD    STRESSES  01 


f  - 


Shear  in  intermediate  panel;   general  case. 

:        '       S^M,_M,       M,_M,    ' 
L         p          p         L 

dSb_Wt      W,       W,      TF, 

"^  ==  T"    y  •  •  y  "  T 

Shear  in  intermediate  panel;  usual  case. 


.  7.     Through  Pratt  Truss  with  Inclined  Chord. 

Stress  in  hanger.     Use  formulas  (14a)  and  (15a). 
Stress  in  any  horizontal  chord  member;  usual  case. 


Compression  in  any  inclined  top  chord  member  or  end 
post;  usual  case. 


.  (22) 

Compression  in  vertical  post;   usual  case. 

(28) 

Stresses  in  inclined  web  members  including  counters; 
usual  case. 


v 


..-  St,  &,  S,  =  M4  -          Mt    .   .    .   (24) 

Stress  in  inclined  counter;    special  case  of  loading  ad- 
anced bevond  panel. 


62  LIVE-LOAD    STRESSES 

/  ta  \  ,  t  ( ,f       b  ,T  \      (  ta 

S2  =  (-rriMi  —  T-\Mz  —  — MZ)  =  (-rr 
\cbL'  bp\  c       /      \coL 

Counter- tension  in  vertical  post;    usual  case. 


Formulas  (21),  (23),  and  (24)  are  of  the  general  form 

S  =  GM,  -  HM3     •  .    . '.,.  ,  V  .   (27) 
where  the  coefficients  G  and  H  may  be  tabulated  thus: 

Type  of  member G  H 

Horizontal  chord —  - 

Vertical  post ry-  — 

Inclined  web  member. .  .-7-7-  7— 

cbL  bp 

The  rate  of  variation  of  S  in  formula  (27)  is 
dS 


(28) 

7          —     v  r  r    4  J.J.   rr   3     —     J.x   »    -rj  r  r    4  rr    A  I  •       •       \***/ 

When  S  in  formulas  (21),  (23),  or  (24)  is  a  maximum 

CC*  \ 

jjWi  —  Wzj  passes  through  zero. 

Through  Pratt  Truss— Parallel  Chords. 
Stress  in  hanger, — use  formulas  (14a)  and  (15a) 

Stress  in  horizontal  chord  =  S~a  =  \ — jM^  —  \-jM3  ,  (21) 
"  vertical  post  =  &  =  (|)M4—  (-\M*      •    •    (29) 

VJL//  \  p  / 


"  inclined  web  =  S,  =          M*  -    ~^i  =         (30) 


LIVE-LOAD    STRESSES  63 

Stress  in  end  post  =  $6  =  ~Sb      ........    (22) 

Formulas  (21),  (29),  and  (30)  are  of  the  general  form 

S  =  G-  M,  -  H-Mt        .    .    ....   (27) 

and  their  rate  of  variation  is 

.    .    .    .    -(28) 


G  and  H  are  the  coefficients  of  M4  and  Ms  in  equations  (21), 
(29),  and  (30),  respectively. 

When  S  in  formulas  (21),  (29),  or  (30)  is  a  maximum, 

/  C1  \ 

\JfWi  —  W3}  passes  through  zero. 

Art.  9.     Equivalent  Uniform  Loads. 

~~j ^O-Ly 

(32) 


INDEX  TO  TABLES 

NO.  PAGE 

1  Five  standard  loadings 66 

2  Lengths,  loads,  load  sums,  and  moment  sums  from  1  to  400  ft. 

for  five  standard  loadings 67 

3  Position  of  Cooper's  loadings  for  maximum  pier  reaction  between 

equal  and  unequal  beam  spans 88 

4  Position  of  Cooper's  loadings  for  absolute  maximum  bending  mo- 

ment in  beam  spans 89 

5  Position  of  Cooper's  loadings  for  maximum  end  shear  in  beam 

spans 89 

6  Position  of  Cooper's  loadings  for  maximum  panel  shear  in  bridges 

with  equal  panels 90 

7  Maximum  moments,  shears,  and  pier  reactions  for  beam  spans — 

Cooper's  #40  and  #50  loadings 91 

8  Maximum   moments   at   panel   points   of   truss   bridges — Cooper's 

#50  loading 94 

9  Maximum  shears  in  panels  of  truss  bridges.     Cooper's  #50  loading      97 

10  Maximum  bending  moments  at  5-foot  intervals  of  beams — Cooper's 

#40  loading 100 

11  Maximum  bending  moments  at  5-foot  intervals  of  beams — Cooper's 

#50  loading 102 

12  Maximum  bending  moments  at  5-foot  intervals  of  beams — Cooper's 

#60  loading 104 

13  Maximum   pier   reactions   for   equal   and   unequal   beam   spans — 

Cooper's  #40  loading 106 

14  Maximum   pier   reactions   for   equal   and   unequal  beam   spans — 

Cooper's  #50  loading 108 

15  Maximum   pier   reactions   for   equal   and   unequal   beam   spans — 

Cooper's  #60  loading 110 

16  Equivalent  uniform  loads — Cooper's  #40  loading 112 

17  Equivalent  uniform  loads — Cooper's  #50  loading 114 

18  Equivalent  uniform  loads — Cooper's  #60  loading 116 

19  Influence-line  ordinates  for  bending  moments  at  5-foot  intervals  of 

beam  spans 118 

20  Reciprocals  of  influence-line  ordinates  for  bending  moments  at  5-foot 

intervals  of  beam  spans 120 

21  Bending  moments  at  5-foot  intervals  of  beam  spans  due  to  a  unit 

uniform  load  .  122 


65 


LIVE-LOAD    STRESSES 

TABLE  1 

STANDARD  LOADINGS 
Loads  given  are  for  one  rail. 


COOPER'S  E  40: 


COOPER'S  E  50: 


111 

(jXJXJxt)  cj>  c|)  cL  cj>  4  fofcb6  ^cLo  F 

'ieis^sj 


COOPER'S  E  60: 


SC50CS          «0  OOQS  C50SC50S 

T-HT-IT--I  -i  CCCOCOCO  rH       rH     -H      i— I 


^o 


OO    I    3000  I bs.  per  ft. 


8H*-«M««JMW«6H^H*W*6WWii 


COMMON  STANDARD-1904-PACIF1C  SYSTEM 


D.  L.  &  W.  R.  R.: 


LIVE-LOAD    STRESSES  67 


TABLE  2 

LOAD  SUMS  AND  MOMENT  SUMS  FOR  COOPER'S 
AND  OTHER  STANDARD  LOADINGS 

NOTE. — Load  Sums  and  Moment  Sums  are  given  per  rail  in  thousands 
of  pounds   and   foot-pounds  respectively. 


68  LIVE-LOAD    STRESSES 

COOPER'S  #40.     0'-50'  COOPER'S  #40.     50'-100' 


Length 

Wheel 

Load 

Load 
Sums 

Moment 
Sums 

Length 

Wheel 

Load 

Load 

Sums 

Moment 
Sums 

w  1 

10 

10 

o 

50 

3780 

i 

W  •  J. 

10 

51 

3922 

J. 

2 

20 

52 



4064 

3 

30 

53 



4206 

4 

40 

54 

4348 

5 

50 

55 



4490 

6 

t 

60 

56 

w.  10 

io 

152 

4632 

7 

70 

57 

4784 

8 

w.  2 

20 

'30 

80 

58 



. 

4936 

9 

110 

59 

5088 

10 

140 

60 

5240 

11 

170 

61 



5392 

12 

200 

62 



5544 

13 

w.'3 

20 

'so 

230 

63 

5696 

14 

280 

64 

w.  ii 

20 

172 

5848 

15 

330 

65 

6020 

16 

380 

66 

6192 

17 

430 

67 

6364 

1C 

w  4 

20 

70 

480 

68 

6536 

J.O 

19 

550 

69 

w.  12 

20 

i92 

6708 

20 

620 

70 



6900 

21 

690 

71 

7092 

22 

760 

72 

7284 

23 

w.'5 

20 

'90 

830 

73 

7476 

24 

920 

74 

w.  13 

20 

212 

7668 

25 

1010 

75 

7880 

26 

.... 

1100 

76 



8092 

27 

1190 

77 



8304 

28 

1280 

78 



8516 

29 

1370 

79 

w.  14 

20 

232 

8728 

30 

1460 

80 

;-_ 

8960 

31 

1550 

81 



9192 

32 

w.  6 

is 

103 

1640 

82 

9424 

33 

.... 

1743 

83 

.  .  . 

9656 

34 

.... 

1846 

84 

9888 

35 

1949 

85 

*  . 

10120 

36 

2052 

86 

10352 

37 

w.  7 

13 

116 

2155 

87 

10584 

38 

2271 

88 

w.  15 

13 

245 

10816 

39 

2387 

89 

11061 

40 

2503 

90 

11306 

41 

2619 

91 

11551 

42 

2735 

92 

11796 

43 

w.8 

13 

129 

2851 

93 

w.  16 

13 

258 

12041 

44 

2980 

94 

12299 

45 

3109 

95 

12557 

46 

3238 

96 

.  .  . 

12815 

47 

.... 

3367 

97 

.  . 

.  .  . 

13073 

48 

Sv.9 

is 

i42 

3496 

98 

13331 

49 

3638 

99 

w.  17 

is 

27i 

13589 

50 

3780 

100 

•  •> 

13860 

LIVE-LOAD    STRESSES  69 

COOPER'S  #40.     100'-150'  COOPER'S  #40.     150'-200' 


Length 

Wheel 

Load 

Load 

Sums 

Moment 
Sums 

Length 

Load 

Load 
Sums 

Moment 
Sums 

100 

13860 

150 

366 

29689 

101 

14131 

151 

368 

30056 

102 

103 



14402 
14673 

152 
153 

370 
372 

30425 
30796 

104 
105 

W.  18 

13 

284 

14944 
15228 

154 
155 

374 
376 

31169 
31544 

106 

15512 

156 

378 

31921 

107 

15796 

157 

380 

32300 

108 

16080 

158 

382 

32681 

109 

284 

16364 

159 

384 

33064 

110 

286 

16649 

160 

386 

33449 

111 

288 

16936 

161 

388 

33836 

112 

290 

17225 

162 

390 

34225 

113 
114 
115 
116 



292 
294 
296 
298 

17516 
17809 
18104 
18401 

163 
164 
165 
166 

392 
394 
396 
398 

34616 
35009 
35404 
35801 

117 

300 

18700 

167 

400 

36200 

118 

302 

19001 

168 

402 

36601 

119 

304 

19304 

169 

404 

37004 

120 
121 

1 

f_! 

306 
308 

19609 
19916 

170 
171 

I 

JH 

406 
408 

37409 
37816 

122 

& 

310 

20225 

172 

OH 

410 

38225 

123 

•a 

312 

20536 

173 

02 

412 

38636 

124 
125 

314 
316 

20849 
21164 

174 
175 

1 

o 

414 
416 

39049 
39464 

126 

318 

21481 

176 

a 

418 

39881 

127 

§ 

320 

21800 

177 

420 

40300 

128 

f\T 

322 

22121 

178 

422 

40721 

129 

130 
131 
132 

:!:| 

324 

326 

328 
330 

22444 

22769 
23096 
23425 

179 

180 
181 

182 

<N 
II 

424 

426 

428 
430 

41144 

41569 
41996 
42425 

133 

& 

332 

23756 

183 

| 

432 

42856 

134 
135 
136 
137 
138 
139 

140 
141 
142 
143 

4 

a 

P 

334 
336 
338 
340 
342 
344 

346 
348 
350 
352 

24089 
24424 
24761 
25100 
25441 
25784 

26129 
26476 

26825 
27176 

184 
185 
186 

187 
188 
189 

190 
191 
192 
193 

| 

'2 
P 

434 
436 
438 
440 
442 
444 

446 
448 
450 
452 

43289 
43724 
44161 
44600 
45041 
45484 

45929 
46376 
46825 
47276 

144 
145 
146 
147 
148 
149 
150 

354 
356 
358 
360 
362 
364 
366 

27529 

27884 
28241 
28600 
28961 
29324 
29689 

194 
195 
196 
197 
198 
199 
200 

454 
456 
458 
460 
462 
464 
466 

47729 
48184 
48641 
49100 
49561 
50024 
50489 

70 


LIVE-LOAD    STRESSES 


COOPER'S  #40.     200'-250' 


COOPER'S  #40.     250-300' 


Length 

Load 

Load 

Sums 

Moment 
Sums 

Length 

Load 

Load 

Sums 

Moment 
Sums 

200 

466 

50489 

250 

566 

76289 

201 

468 

50956 

251 

568 

76856 

202 

470 

51425 

252 

570 

77425 

203 

472 

51896 

253 

572 

77996 

204 

474 

52369 

254 

574 

78569 

205 

476 

52844 

255 

576 

79144 

206 

478 

53321 

256 

578 

79721 

207 

480 

53800 

257 

580 

80300 

208 

482 

54281 

258 

582 

80881 

209 

484 

54764 

259 

584 

81464 

210 

486 

55249 

260 

586 

82049 

211 

488 

55736 

261 

588 

82636 

212 

490 

56225 

262 

590 

83225 

213 

492 

56716 

263 

592 

83816 

214 

494 

57209 

264 

594 

84409 

215 

496 

57704 

265 

596 

85004 

216 

.  498 

58201 

266 

598 

85601 

217 

* 

500 

58700 

267 

+3 

600 

86200 

218 

J 

502 

59201 

268 

602 

86801 

219 

& 

504 

59704 

269 

& 

604 

87404 

220 

£ 

506 

60209 

270 

.3 

606 

88009 

221 

1 

508 

60716 

271 

1 

608 

88616 

222 

510 

61225 

272 

a"* 

610 

89225 

223 

o, 

512 

61736 

273 

612 

89836 

224 

514 

62249 

274 

614 

90449 

225 

516 

62764 

275 

616 

91064 

226 

of 

518 

63281 

276 

oC 

618 

91681 

227 

1! 

520 

63800 

277 

II 

620 

92300 

228 

•a 

522 

64321 

278 

*s 

622 

92921 

229 

J 

524 

64844 

279 

3 

624 

93544 

230 

a 

526 

65369 

280 

e 

626 

94169 

231 

!_ 
£ 

528 

65896 

281 

c 
M 

628 

94796 

232 

3 

530 

66425 

282 

*3 

g 

630 

95425 

233 

6 

532 

66956 

283 

p 

632 

96056 

234 

534 

67489 

284 

634 

96689 

235 

536 

68024 

285 

636 

97324 

236 

538 

68561 

286 

638 

97961 

237 

540 

69100 

287 

640 

98600 

238 

542 

69641 

288 

642 

99241 

239 

544 

70184 

289 

644 

99884 

240 

546 

70729 

290 

646 

100529 

241 

548 

71276 

291 

648 

101176 

242 

550 

71825 

292 

650 

101825 

243 

552 

72376 

293 

652 

102476 

244 

554 

72929 

294 

654 

103129 

245 

556 

73484 

295 

656 

103784 

246 

558 

74041 

296 

658 

104441 

247 

560 

74600 

297 

660 

105100 

248 

562 

75161 

298 

662 

105761 

249 

564 

75724 

299 

664 

106424 

250 

566 

76289 

300 

606 

107089 

LIVE-LOAD    STRESSES  71 

COOPER'S  #40.     300'-350'  COOPER'S  £40.     350'-400' 


Length 

Load 

Load 
Sums 

Moment 
Sums 

Length 

Load 

Load 
Sums 

Moment 
Sums 

300 

666 

107089 

350 

766 

142889 

30J 

668 

107756 

351 

768 

143656 

302 

670 

108425 

352 

770 

144425 

303 

672 

109096 

353 

772  • 

145196 

304 

674 

109769 

354 

774 

145969 

305 

676 

110444 

355 

776 

146744 

306 

678 

111121 

356 

778 

147521 

307 

680 

111800 

357 

780 

148300 

308 

682 

112481 

358 

782 

149081 

309 

684 

113164 

359 

784 

149864 

310 

686 

113849 

360 

786 

150649 

311 

688 

114536 

361 

788 

151436 

312 

690 

115225 

362 

790 

152225 

313 

692 

115916 

363 

792 

153016 

314 

694 

116609 

364 

794 

153809 

315 

696 

117304 

365 

796 

154604 

316 

698 

118001 

366 

798 

155401 

317 

"o 

700 

118700 

367 

Q 

800 

156200 

318 

8 

702 

119401 

368 

1 

802 

157001 

319 

Lj 

704 

120104 

369 

IH 

804 

157804 

PH 

& 

320 

00 

706 

120809 

370 

go 

806 

158609 

321 

a 

708 

121516 

371 

-d 

G 

808 

159416 

322 

710 

122225 

372 

P 

810 

160225 

323 

712 

122936 

373 

812 

161036 

324 

714 

123649 

374 

814 

161849 

325 

716 

124364 

375 

816 

162664 

326 

c* 

718 

125081 

376 

eC 

818 

163481 

327 

II 

720 

125800 

377 

II 

820 

164300 

328 

cr? 

722 

126521 

378 

"d 

822 

165121 

329 

1 

724 

127244 

379 

J 

824 

165944 

330 

726 

127969 

380 

a 

826 

166769 

331 

0 

728 

128696 

381 

1 

828 

167596 

332 

a 

Q 

730 

129425 

382 

2 

830 

168425 

333 

P 

732 

130156 

383 

832 

169256 

334 

734 

130889 

384 

834 

170089 

335 

738 

131624 

385 

836 

170924 

336 

738 

132361 

386 

838 

171761 

337 

•  740 

133100 

387 

840 

172600 

338 

742 

133841 

388 

842 

173441 

339 

744 

134584 

389 

844 

174284 

340 

746 

135329 

390 

846 

175129 

341 

748 

136076 

391 

848 

175976 

342 

750 

136825 

392 

850 

176825 

343 

752 

137576 

393 

852 

177676 

344 

754 

138329 

394 

854 

178529 

345 

756 

139084 

395 

856 

179384 

346 

758 

139841 

396 

858 

180241 

347 

760 

140600 

397 

860 

181100 

348 

762 

141361 

398 

862 

181961 

349 

764 

142124 

399 

864 

182824 

350 

766 

142889 

400 

866 

183689 

72 


LIVE-LOAD    STRESSES 


COOPER'S  ESQ.     0'-50' 


COOPER'S  #50.     50'-100' 


Length 

Wheel 

Load 

Load 
Sums 

Moment 
Sums 

Length 

Wheel 

Load 

Load 
Sums 

Moment 
Sums 

o 

W.  1 

12.50 

12  50 

00.00 

50 

4725  00 

1 

12.50 

51 

4902  50 

2 

25.00 

52 

5080  00 

3 

37.50 

53 

5257  50 

4 

50.00 

54 

5435  00 

5 

62.50 

55 

5612  50 

6 

75.00 

56 

w.  10 

1250 

190  00 

5790  00 

7 

87.50 

57 

5980  00 

8 

w.  2 

25.00 

37.50 

100.00 

58 

6170  00 

9 

137.50 

59 

636000 

10 

175.00 

60 

6550  00 

11 

212.50 

61 

6740  00 

12 

250.00 

62 

6930  00 

13 

w.  3 

25.00 

62.50 

287.50 

63 

7120  00 

14 

350.00 

64 

w.  11 

25.00 

215  00 

7310  00 

15 

412.50 

65 

7525  00 

16 

475.00 

66 

7740  00 

17 

537.50 

67 

7955  00 

18 

w.  4 

25.00 

87.50 

600.00  ! 

68 

8170  00 

19 

687.50 

69 

w.  12 

25.00 

240  66 

8385  00 

20 

775.00 

70 

8625  00 

21 

862.50 

71 

8865  00 

22 

950.00 

72 

9105  00 

33 

w.  5 

25.00 

112.50 

1037.50 

73 

9345  00 

24 

1150.00 

74 

w.  13 

25.00 

265  00 

9585.00 

25 

1262  50 

75 

9850  00 

26 

1375.00 

76 

10115  00 

27 

1487.50 

77 

10380  00 

28 

1600.00 

78 

10645  00 

29 

1712.50 

79 

w.  14 

25.66 

290  00 

10910  00 

30 

1825.00 

80 

1120000 

31 

1937.50 

81 

1149000 

32 

w.  6 

16.25 

128.75 

2050.00 

82 

1178000 

33 

2178.75 

83 

12070  00 

34 

2307.50 

84 

12360.00 

35 

2436  25 

85 

12650  00 

36 

2565.00 

86 

12940.00 

37 

w.  7 

16.25 

145.00 

2693.75 

87 

• 

13230.00 

38 

2838.75 

88 

w  15 

1625 

306  25 

13520  00 

39 

2983.75 

89 

13826.25 

40 

3128.75 

90 

14132  50 

41 

3273.75 

91 

14438  75 

42 

3418.75 

92 

14745.00 

43 

44 

w.  8 

16.25 

161.25 

3563.75 
3725.00 

93 
94 

w.  16 

16.25 

322.50 

15051.25 
15373.75 

45 

3886  25 

95 

15696  25 

46 

4047  50 

96 

16018  75 

47 

4208  75 

97 

16341  25 

48 

w.  9 

16.25 

177.50 

4370.00 

98 

16663.75 

49 

50 

4547.50 
4725.00 

99 
100 

w.  17 

16.25 

338.75 

16986.25 
17325.00 

LIVE-LOAD    STRESSES  73 

COOPER'S  #50.     100'-150'  COOPER'S  #50.     ISO7- 200' 


Length 

Wheel 

Load 

Load 

Sums 

Moment 
Sums 

Length 

Load 

Load 
.  Sums 

Moment 
Sums 

100 

173?5  .  00 

150 

457.50 

37111.25 

101 

17663.75 

151 

460.00 

37570.00 

102 

18002.50 

152 

462.50 

38031.25 

103 

18341.25 

153 

465.00 

38495.00 

104 
105 

w.  18 

16.25 

355.00 

18680.00 
19035  00 

154 
155 

467.50 
470  00 

38961.25 
39430  00 

106 

19390  00 

156 

472  50 

39901  25 

107 

19745  00 

157 

475  00 

40375  00 

108 

20100  00 

158 

477  .  50 

40851  25 

109 

355.00 

20455  00 

159 

480  .  00 

41330.00 

110 
111 
112 
113 
114 



357.50 
360.00 
362.50 
365.00 
367.50 

20811.25 
21170.00 
21531:25 
21895.00 
22261  25 

160 
161 
162 
163 
164 

482.50 
485.00 
487.50 
490.00 
492  50 

41811.25 
42295.00 
42781.25 
43270.00 
43761  25 

115 

370  .  00 

22630  00 

165 

495  00 

44255.00 

116 

372  .  50 

23001  25 

166 

497  50 

44751.25 

117 

| 

375  .  00 

23375  00 

167 

0 

500.00 

45250.00 

118 

5 

377.50 

23751.25 

168 

& 

502  .  50 

45751.25 

119 

£ 

380.00 

24130.00 

169 

ft 

505.00 

46255.00 

120 

A 
•9 

382  50 

24511  25 

170 

s, 

JS 

507  50 

46761  25 

121 
122 
123 
124 
125 



1 

rvT 

385.00 
387.50 
390.00 
392.50 
395  00 

24895.00 
25281.25 
25670.00 
26061.25 
26455  00 

171 
172 
173 
174 
175 

1 

510.00 
512.50 
515.00 
517.50 
520  00 

47270.00 
47781.25 
48295.00 
48811.25 
49330  00 

126 

397  50 

26851  25 

176 

<N 

522  50 

49851.25 

127 

" 

400  .  00 

27250  00 

177 

II 

525  .  00 

50375.00 

128 
129 

130 
131 
132 
133 



Uniform  Loac 

402.50 
405.00 

407.50 
410.00 
412.50 
415.00 

27651.25 
28055.00 

28461.25 
28870.00 
29281.25 
29695  .  00 

178 
179 

180 
181 
182 
183 

Uniform  Loac 

527.50 
530.00 

532.50 
535.00 
537.50 
540  00 

50901.25 
51430.00 

51961.25 
52495.00 
53031.25 
53570.00 

134 

417.50 

30111.25 

184 

542  50 

54111.25 

135 
136 

420.00 
422.50 

30530.00 
30951.25 

185 

186 

545.00 
547  .  50 

54655.00 
55201.25 

137 
138 
139 

140 
141 

425.00 
427.50 
430.00 

432.50 
435  00 

31375.00 
31801.25 
32230.00 

32661.25 
33095  00 

187 
188 
189 

190 
191 

550.00 
552.50 
555.00 

557.50 
560  00 

55750.00 
56301.25 
56855.00 

57411.25 
57970.00 

142 

437  50 

33531  25 

192 

562  50 

58531.25 

143 

440  00 

33970  00 

193 

565.00 

59095  .  00 

144 
145 
146 
147 
148 

442.50 
445.00 
447.50 
450.00 
452  .  50 

34411.00 
34855.00 
35301.25 
35750.00 
36201  .  25 

194 
195 
196 
197 
198 

567.50 
570.00 
572.50 
575.00 
577.50 

59661.25 
60230.00 
60801.25 
61375.00 
61951.25 

149 
150 



455.00 
457.50 

36655.00 
37111.25 

199 
200 

580.00 
582.50 

62530.00 
63111.25 

74 


LIVE-LOAD    STRESSES 


COOPER'S  £50.     200 '-250' 


COOPER'S  #50.     250 '-300 


Length 

Load 

Load       Moment 
Sums        Sums 

Length 

Load 

Load 
Sums 

Moment 
Sums 

200 

582.50 

63111.25 

250 

707.50 

95361.25 

201 

585.00 

63695.00 

251 

710.00 

96070.00 

202 

587.50 

64281.25 

252 

712.50 

96781.25 

203 

590.00 

64870.00 

253 

715.00 

97495.00 

204 

592.50 

65461.25 

254 

717.50 

98211.25 

205 

595.00 

66055.00 

255 

720.00 

98930.00 

206 

597.50 

66651.25 

256 

722.50 

99651.25 

207 

600.00 

67250.00 

257 

725.00 

100375.00 

208 

602.50 

67851.25 

258 

727.50 

101101.25 

209 

605.00 

68455.00 

259 

730.00 

101830.00 

210 

607.50 

69061.25 

260 

732.50 

102561.25 

211 

610.00 

69670.00 

261 

735.00 

103295.00 

212 

612.50 

70281.25 

262 

737.50 

104031.25 

213 

615.00 

70895.00 

263 

740.00 

104770.00 

214 

617.50 

71511.25  , 

264 

742.50 

105511.25 

215 

620.00 

72130.00 

265 

745.00 

106255.00 

216 

622.50 

72751.25 

266 

747.50 

107001.25 

217 

+3 

625.00 

73375.00 

267 

49 

750.00 

107750.00 

218 

§ 

627.50 

74001.25   268 

1 

752.50 

108501.25 

219 

£+ 

630.00 

74630.00 

269 

• 

755.00 

109255.00 

220 

I 

632.50 

75261.25 

270 

I 

757.50 

110011.25 

221 

3 

635.00 

75895.00 

271 

•o 

760.00 

110770.00 

222 

B 

637.50 

76531.25 

272 

d 

3 

762.50 

111531.25 

223 

a 

640.00 

77170.00 

273 

a 

765.00 

112295.00 

224 

642.50 

77811.25 

274 

767.50 

113061.25 

225 

g 

645.00 

78455.00 

275 

s 

770.00 

113830.00 

226 

c<f 

647.50 

79101.25 

276 

eC 

772.50 

114601.25 

227 

II 

650.00 

79750.00 

277 

II 

775.00 

115375.00 

228 

T3 

652.50 

80401.25 

278 

...j 

777.50 

116151.25 

229 

0 

655.00 

81055.00 

279 

0 

780.00 

116930.00 

230 

fl 

657.50 

81711.25 

280 

^ 

782.50 

117711.25 

231 

§ 

660.00 

82370.00 

281 

Q 

785.00 

118495.00 

232 

| 

662.50 

83031.25 

282 

*§ 

787.50 

119281.25 

233 

I 

665.00 

83695.00 

283 

G 
|~i 

790.00 

120070.00 

234 

h-' 

667.50 

84361.25 

284 

792.50 

120861.25 

235 

670.00 

85030.00 

285 

795.00 

121655.00 

236 

672.50 

85701.25 

286 

797.50 

122451.25 

237 

675.00 

86375.00 

287 

800.00 

123250.00 

238 

677.50 

87051.25 

288 

802.50 

124051.25 

239 

680.00 

87730.00 

289 

805.00 

124855.00 

240 

682.50 

88411.25 

290 

807.50 

125661.25 

241 

685.00 

89095.00 

291 

810.00 

126470.00 

242 

687.50 

89781.25   292 

812.50 

127281.25 

243 

690.00 

90470.00   293 

815.00   128095.00 

244 

692.50 

91161.25  i  294 

817.50   128911.25 

245 

695.00 

91855.00   295        820.00   129730.00 

246 

697.50    92551.25 

296 

822.50   130551.25 

247 

700.00  •  93250.00 

297 

825.00  i  131375.00 

248 

702.50    93951.25 

298 

827.50   132201.25 

249 

705.00    94655.00 

299 

830.00  ;  133030.00 

250 

707.50  i  95361.25 

300 

832.50  i  133861.25 

LIVE-LOAD    STRESSES 


75 


COOPER'S  #50.     3()0'-350' 


COOPER'S  #50.     350'-400' 


Length 

Load 

Load 
Sums 

Moment 
Sums 

Length 

Load 

Load      Moment 
Sums        Sums 

300 

832.50 

133861.25 

350 

957.50 

178611.25 

301 

835.00 

134695.00 

351 

960.00 

179570.00 

302 

837.50 

135531.25 

352 

962.50 

180531.25 

303 

840.00 

136370.00 

353 

965.00 

181495.00 

304 

842.50 

137211.25 

354 

967.50 

182461.25 

305 

845.00 

138055.00 

355 

970.00 

183430.00 

306 

847.50 

138901.25 

356 

972.50 

184401.25 

307 

850.00 

139750.00 

357 

975.00 

185375.00 

308 

852.50 

140601.25 

358 

977.50 

186351.25 

309 

855.00 

141455.00 

359 

980.00 

187330.00 

310 

857.50 

142311.25 

360 

982.50 

188311.25 

311 

860.00 

143170.00 

361 

985.00 

189295.00 

312 

862.50 

144031.25 

362 

987.50 

190281.25 

313 

865.00 

144895.00 

363 

990.00 

191270.00 

314 

867.50 

145761.25 

364 

992.50 

192261.25 

315 

870.00 

146630.00 

365 

995.00 

193255.00 

316 

872.50 

147501.25 

366 

997.50 

194251.25 

317 

40 

875.00 

148375.00 

367 

,u 

1000.00 

195250.00 

318 

8 

877.50 

149251.25 

368 

8 

1002.50 

196251.25 

319 

t+H 

880.00 

150130.00 

369 

*M 

5^ 

1005.00 

197255.00 

320 

I 

882.50 

151011.25 

370 

1 

1007.50 

198261.25 

321 

| 

885.00 

151895.00 

371 

1 

1010.00 

199270.00 

322 

£j 

887.50 

152781.25 

372 

Q 

13 

1012.50 

200281.25 

323 

a 

890.00 

153670.00 

373 

a 

1015.00 

201295.00 

324 

892.50 

154561.25 

374 

1017.50 

202311.25 

325 

o 

895.00 

155455.00 

375 

§ 

1020.00 

203330.00 

326 

<N~ 

897.50 

156351.25 

376 

cC 

1022.50 

204351.25 

327 

|| 

900.00 

157250.00 

377 

II 

1025.00 

205375.00 

328 

T3 

902.50 

158151.25 

378 

T5 

1027.50 

206401.25 

329 

2 

905.00 

159055.00 

379 

j 

1030.00 

207430.00 

330 

£ 

907.50 

159961.25 

380 

1032.50 

208461.25 

331 

s 

910.00 

160870.00 

381 

c 

1035.00 

209495.00 

332 

£ 

912.50 

161781.25 

382 

J§ 

1037.50 

210531.25 

333 

1 

915.00 

162695.00 

383 

C5 
\~  j 

1040.00 

211570.00 

334 

H-* 

917.50 

163611.25 

384 

H^ 

1042.50 

212611.25 

335 

920.00 

164530.00 

385 

1045.00 

213655.00 

336 

922.50 

165451.25 

386 

1047.50 

214701.25 

337 

925.00 

166375.00 

387 

1050.00 

215750.00 

338 

927.50 

167301.25 

388 

1052.50 

216801.25 

339 

930.00 

168230.00 

389 

1055.00 

217855.00 

340 

932.50 

169161.25 

390 

1057.50 

218911.25 

341 

935.00 

170095.00 

391 

1060.00 

219970.00 

342 

937.50 

171031.25 

392 

1062.50 

221031.25 

343 

940.00 

171970.00 

393 

1065.00 

222095.00 

344 

942.50 

172911.25 

394 

1067.50 

223161.25 

345 

945.00 

173855.00 

395 

1070.00 

224230.00 

346 

947.50 

174801.25 

396 

1072.50 

225301.25 

347 

950.00 

175750.00 

397 

1075.00 

226375.00 

348 

952.50 

176701.25 

398 

1077.50 

227451.25 

349 

955.00 

177655.00 

399 

1080.00 

228530.00 

350 

957.50 

178611.25 

400 

1082.50 

229611.25 

76  LIVE-LOAD    STRESSES 

COOPER'S  #60.     0'-50'  COOPER'S  #60.     50'-100' 


Length 

Wheel 

Load 

Load 
Sums 

Moment 
Sums 

Length 

Wheel 

Load 

Load 
Sums 

Moment 
Sums 

o 

W  1 

15  0 

15.0 

00.00 

50 

5670  00 

1 

15.00 

51 

5883  00 

2 

30  00 

52 

6096  00 

3 

45  00 

53 

6309  00 

4 

60  00 

54 

6522  00 

5 

75  00 

55 

6735  00 

6 

90.00 

56 

w.  10 

15  0 

228.0 

6948  00 

7 

105.00 

57 

7176  00 

8 

w.  2 

30.0 

45.0 

120.00 

58 

7404  .  00 

9 

165  00 

59 

7632  00 

10 

210.00 

60 

7860  00 

11 

255.00 

61 

8088  .  00 

12 

300  00 

62 

8316  00 

13 

w  3 

30  0 

75.0 

345  00 

63 

8544  00 

14 

420.00 

64 

w.  11 

30.0 

258.0 

8772.00 

15 

495  .  00 

65 

9030  00 

16 

570.00 

66 

9288  00 

17 

645.00 

67 

9546  .  00 

18 
19 

w.  4 

30.0 

105.0 

720.00 
825.00 

68 
69 

w.  12 

30.6 

288.6 

9804.00 
10062.00 

20 

930.00 

70 

10350.00 

21 

1035  00 

71 

10638  00 

22 

1140  00 

72 

10926  00 

23 

w  5 

30  0 

135.0 

1245  00 

73 

11214  00 

24 

1380  00 

74 

w.  13 

30  0 

318.0 

11502  00 

25 

1515.00 

75 

11820.00 

26 

1650.00 

76 

12138.00 

27 

1785.00 

77 

12456.00 

28 

1920  00 

78 

12774  00 

29 

2055.00 

79 

w.  14 

30.0 

348.0 

13092.00 

30 

2190.00 

80 

13440.00 

31 

2325  00 

81 

13788  00 

32 

w  6 

19  5 

1545 

2460  00 

82 

14136  00 

33 

2614  50 

83 

14484  .  00 

34 

2769  00 

84 

14832.00 

35 

2923  .  50 

85 

15180.00 

36 

3078  .  00 

86 

15528.00 

37 

w  7 

19  5 

174  0 

3232  50 

87 

15876  00 

38 

3406.50 

88 

w.  15 

19.5 

367.5 

16224.00 

39 

3580  50 

89 

16591.00 

40 

3754  50 

90 

16959  00 

41 

3928  50 

91 

17326.50 

42 

4102  50 

92 

17694.00 

43 
44 

w.  8 

19.5 

193.5 

4276.50 
4470  00 

93 
94 

w.  16 

19.5 

387.0 

18061.50 
18448.00 

45 

4663  50 

95 

18835  50 

46 

4857  00 

96 

19222  .  50 

47 

5050  50 

97 

19609.50 

48 

w.  9 

19  5 

213.0 

5244  00 

98 

19996.50 

49 

5457  00 

99 

w.  17 

19.5 

406.5 

20383.50 

50 

5670  00 

100 

20790.00 

LIVE-LOAD    STRESSES 


77 


COOPER'S  #60.     100'-150' 


COOPER'S  #60.     150'-200' 


Length 

Wheel 

Load 

Load 
Sums 

Moment 
Sums 

Length 

Load 

Load 
Sums 

Moment 
Sums 

100 

20790  .  00 

150 

549.0 

44533  50 

101 

21196.50 

151 

552  '.  0 

45084  .  00 

102 

21603.00 

152 

555.0 

45637  .  50 

103 

22009  50 

153 

558  0 

46194  00 

104 

w.  18 

19.5 

426.0 

22416.00 

154 

561.0 

46753^50 

105 

22842.00 

155 

564.0 

47316.00 

106 

23268  .  00 

156 

567.0 

47881  .  50 

107 

23694  .  00 

157 

570.0 

48450  .  00 

108 

24120.00 

158 

573.0 

4902l!  50 

109 

426^0 

24546'  00 

159 

576  .'O 

49596.'  00 

110 

429.0 

24973.50 

160 

579.0 

50173.50 

111 

432.0 

25404.00 

161 

582.0 

50754.00 

112 

435.0 

25837.50 

162 

585.0 

51337.50 

113 

438.0 

26274.00 

163 

588.0 

51924.00 

114 

441.0 

26713.50 

164 

591.0 

52513.50 

115 

444.0 

27156.00 

165 

594.0 

531.06.00 

116 

447.0 

27601.50 

166 

597.0 

53701.50 

117 

450.0 

28050.00 

167 

600.0 

54300.00 

118 

453.0 

28501.50 

168 

3 

603.0 

54901.50 

119 

456.0 

28956.00 

169 

^ 

606.0 

55506.00 

120 

| 

459.0 

29413.50 

170 

o 

609.0 

56113.50 

121 

462.0 

29874.00 

171 

GO 

612.0 

56724.00 

122 

&c 

465.0 

30337.50 

172 

£ 

615.0 

57337.50 

123 

02 

468.0 

30804.00 

173 

O 

618.0 

57954.00 

124 

C 

471.0 

31273.50 

174 

CX 

621.0 

58573.50 

125 

474.0 

31746.00 

175 

624.0 

59196.00 

126 

a 

477.0 

32221.50 

176 

„ 

627.0 

59821.50 

127 

480.0 

32700.00 

177 

CO 

630.0 

60450.00 

128 

483.0 

33181.50 

178 

" 

633.0 

61081.50 

129 

CO 

486.0 

33666.00 

179 

1 

636.0 

61716.00 

130 

- 

489.0 

34153.50 

180 

3 

639.0 

62353.50 

131 

o 

492.0 

34644.00 

181 

B 

642.0 

62994.00 

132 

i-3 

495.0 

35137.50 

182 

1 

645.0 

63637.50 

133 

g 

498.0 

35634.00 

183 

'3 

648.0 

64284.00 

134 

o 

501.0 

36133.50 

184 

P 

651.0 

64933.50 

135 

^2 

504.0 

36636.00 

185 

654.0 

65586.00 

136 

p 

507.0 

37141.50 

186 

657.0 

66241.50 

137 

510.0 

37650.00 

187 

660.0 

66900.00 

138 

513.0 

38161.50 

188 

663.0 

67561.50 

139 

516.0 

38676.00 

189 

666.0 

68226.00 

140 

519.0 

39193.50 

190 

669.0 

68893.50 

141 

522.0 

39714.00 

191 

672.0 

69564.00 

142 

525.0 

40237.50 

192 

675.0 

70237.50 

143 

528.0 

40764.00 

193 

678.0 

70914.00 

144 

531.0 

41293.50 

194 

681.0 

71593.50 

145 

534.0 

41826.00 

195 

684.0 

72276.00 

146 

537.0 

42361.50 

196 

687.0 

72961.50 

147 

540.0 

42900.00 

197 

690.0 

73650.00 

148 

543.0 

43441.50 

198 

693.0 

74341.50 

149 

546.0 

43986.00 

199 

696.0 

75036.00 

150 

549.0 

44533.50 

200 

699.0 

75733.50 

78  LIVE-LOAD    STRESSES 

COOPER'S  #60.     200'-250'  COOPER'S  ESQ.     250/-300/ 


Length 

Load 

Load 

Sums 

Moment 
Sums 

,  Length 

Load 

Load     Moment 
Sums      Sums 

200 

699.0 

75733.50 

250 

849.0   114433.50 

201 

702.0 

76434.00 

251 

852.0   115284.00 

202 

705.0 

77137.50 

252 

855.0 

116137.50 

203 

708.0 

77844.00 

253 

858.0 

116994.00 

204 

711.0 

78553.50 

254 

861.0 

117853.50 

205 

714.0 

79266.00 

255 

864.0 

118716.00 

206 

717.0 

79981.50 

256 

867.0 

119581.50 

207 

720.0 

80700.00 

257 

870.0 

120450.00 

208 

723.0 

81421.50 

258 

873.0 

121321.50 

209 

726.0 

82146.00 

259 

876.0 

122196.00 

210 

729.0 

82873.50 

260 

879.0 

123073.50 

211 

732.0 

83604.00 

261 

882.0 

123954.00 

212 

735.0 

84337.50 

262 

885.0 

124837.50 

213 

738.0 

85074.00 

263 

888.0 

125724.00 

214 

741.0 

85813.50 

264 

891.0 

126613.50 

215 

744.0 

86556.00 

265 

894.0 

127506.00 

216 

747.0 

87301.50 

266 

897.0 

128401  .  50 

217 

•g 

750.0 

88050.00 

267 

-g 

900.0 

129300.00 

218 

1 

753.0 

88801.50 

268 

1 

903.0 

130201.50 

219 

a 

756.0 

89556.00 

269 

I 

906.0 

131106.00 

220 

& 

759.0 

90313.50 

270 

-SS 

909.0 

132013.50 

221 

73 

762.0 

91074.00 

271 

1 

912.0 

132924.00 

222 

765.0 

91837.50 

272 

915.0 

133837.50 

223 

768.0 

92604.00 

273 

o-    918.0 

134754.00 

224 

771.0 

93373.50 

274 

921.0 

135673.50 

225 

774.0 

94146.00 

275 

924.0 

136596.00 

226 

co" 

777.0 

94921.50 

276 

927.0 

137521.50 

227 

II 

780.0 

95700.00 

277 

»   !  930.0 

138450.00 

228 

•a 

783.0 

96481.50 

278 

-%    933.0 

139381.50 

229 

2 

786.0 

97266.00 

279 

1 

h-1 

936.0 

140316.00 

230 

a 

789.0 

98053.50 

280 

S 

939.0 

141253.50 

231 

o 

792.0 

98844.00 

281 

i 

942.0 

142194.00 

232 

«M 

'2 

795.0 

99637.50 

282 

i 

945.0 

143137.50 

233 

p 

798.0 

100434.00 

283 

948.0 

144084.00 

234 

801.0 

101233.50 

284 

951.0 

145033.50 

235 

804.0 

102036.00 

285 

954.0 

145986.00 

236 

807.0 

102841  .  50 

286 

957.0 

146941.50 

237 

810.0 

103650.00 

287 

960.0 

147900.00 

238 

813.0 

104461.50 

288 

963.0 

148861.50 

239 

816.0 

105276.00 

289 

966.0 

149826.00 

240 

819.0 

106093.50 

290 

969.0 

150793.50 

241 

822.0 

106914.00 

291 

972.0 

151764.00 

242 

825.0 

107737.50 

292 

975.0 

152737.50 

243 

828.0 

108564.00 

293. 

978.0 

153714.00 

244 

831.0 

109393.50 

294 

981.0 

154693.50 

245 

834.0 

110226.00 

295 

984.0 

155676.00 

246 

837.0 

111061.50 

296 

987.0 

156661.50 

247 

840.0 

111900.00 

297 

990.0 

157650.00 

248 

843.0 

112741.50 

298 

993.0 

158641.50 

249 

846.0 

113586.00 

299 

996.0 

159636.00 

250 

849.0 

1144.33.50 

300 

999.0 

160633.50 

LIVE-LOAD    STRESSES 
COOPER'S  #60.     300'-350'  COOPER'S  EGO.     350'-400' 


79 


Length 

Load 

Load 
Sums 

Moment 
Sums 

length 

Load 

Load 
Sums 

Moment 
Sums 

300 

999.0 

160633.50 

350 

1149.0 

214333.50 

301 

1002.0 

161634.00 

351 

1152.0 

215484.00 

302 

1005.0 

162637.50 

352 

1155.0 

216637.50 

303 

1008.0 

163644.00 

353 

1158.0 

217794.00 

304 

1011.0 

164653.50 

354 

1161.0 

218953.50 

305 

1014.0 

165666.00 

355 

1164.0 

220116.00 

306 

1017.0 

166681.50 

356 

1167.0 

221281.50 

307 

1020.0 

167700.00 

357 

1170.0 

222450.00 

308 

1023.0 

168721.50 

358 

1173.0 

223621.50 

309 

1026.0 

169746.00 

359 

1176.0 

224796  .  00 

310 

1029.0 

170773.50 

360 

1179.0 

225973.50 

311 

1032  .  0 

171804.00 

361 

1182.0 

227154.00 

312 

1035.0 

172837.50 

362 

1185.0 

228337.50 

313 

1038.0 

173874.00 

363 

1188.0 

229524.00 

314 

1041.0 

174913.50 

364 

1191.0 

230713.50 

315 

1044.0 

175956.00 

365 

1194.0 

231906.00 

316 

1047.0 

177001.50 

360 

1197.0 

233101.50 

317 

^j 

1050.0 

178050.00 

367 

1200.0 

234300.00 

318 

1 

1053.0 

179101.50 

368 

1 

1203.0 

235501  .  50 

319 

1056.0 

180156.00 

369 

1206.0 

236706.00 

320 

I 

1059.0 

181213.50 

370 

I 

1209  .  0 

237913.50 

321 

•s 

1062.0 

182274.00 

371 

02 

1212.0 

239124.00 

322 

~ 

1085.0 

183337.50 

372 

q 

1215.0 

240337.50 

323 

a 

1038.0 

184404.00 

373 

1 

1218.0 

241554.00 

324 

1071.0 

185473.50 

374 

CM 

1221.0 

242773.50 

325 

g 

1074.0 

186546,00 

375 

0 

1224.0 

243996.00 

326 

w> 

1077.0 

187621.50 

376 

co" 

1227.0 

245221.50 

327 

II 

1030.0 

188700.00 

377 

II 

1230.0 

246450.00 

328 

T3 

1083.0 

189781.50 

378 

1233.0 

247681.50 

329 

a 

1086.0 

190866.00 

379 

1 

1236.0 

248916.00 

330 

d 

1089.0 

191953.50 

380 

w 

1239.0 

250153.50 

331 

I 

1092.0 

193044.00 

381 

3 

1242.0 

251394.00 

332 

<2 

1095.0 

194137.50 

382 

1 

1245.0 

252637.50 

333 

'3 

1098.0 

195234.00 

383 

'd 

1248.0 

253884.00 

334 

^ 

1101.0 

196333.50 

384 

tJ 

1251.0 

255133.50 

335 

1104.0 

197436.00 

385 

1254.0 

256386.00 

336 

1107.0 

198541.50 

386 

1257.0 

257641.50 

337 

1110.0 

199650.00 

387 

1260.0 

258900.00 

338 

1113.0 

200761  .  50 

388 

1263.0 

260161.50 

339 

1116.0 

201876.00 

389 

1266.0 

261426.00 

340 

1119.0 

202993.50 

390 

1269.0 

262693.50 

341 

1122.0 

204114.00 

391 

1272.0 

263964.00 

342  • 

1125.0 

205237.50 

392 

1275.0 

265237.50 

343 

1128.0 

206364.00 

393 

1278.0 

266514.00 

344 

1131.0 

207493.50 

394 

1281.0 

267793.50 

345 

1134.0 

208626.00 

395 

1284.0 

269076.00 

346 

1137.0 

209761.50 

396 

1287.0 

270361.50 

347 

1140.0 

210900.00 

397 

1290.0 

271650.00 

348 

1143.0 

212041.50 

398 

1293.0 

272941.50 

349 

1146.0 

213186.00 

399 

1296.0 

274236.00 

350 

1149.0 

214333.50 

400 

1299.0 

275533  .  50 

80 


LIVE-LOAD    STRESSES 


COMMON  STANDARD  0'-50' 


COMMON  STANDARD  50'-100' 


Length 

Wheel 

Load 

Load 
Sums 

Moment 
Sums 

Length 

. 
Wheel 

Load 

Load 
Sums 

Moment 
Sums 

0 

W  1 

12  5 

12.5 

00.00 

50 

5120.00 

1 

12  50 

51 

5312  50 

2 

25  00 

52 

5505  00 

3 

37  50 

53 

5697.50 

4 

50  00 

54 

5890  .  00 

5 

62  50 

55 

6082.50 

6 

75  00 

56 

W.  10 

12.5 

205.0 

6275.00 

7 

87  50 

57 

6480  00 

8 

W  2 

27  5 

400 

100  00 

58 

6685  .  00 

9 

140.00 

59 

6890.00 

10 

180  00 

60 

7095.00 

11 

220  00 

61 

7300.00 

12 

260  00 

62 

7505.00 

13 
14 

W.  3 

27.5 

67.5 

300.00 
367  .  50 

63 
64 

W.  11 

27.5 

232.5 

7710.00 
7915.00 

15 

435.00 

65 

8147.50 

16 

502  50 

66 

8380.00 

17 

570  00 

67 

8612.50 

18 
19 

w.  4 

27.5 

95.0 

637.50 
732.50 

68 
69 

w.  12 

27.5 

260.0 

8845.00 
9077.50 

20 

827  50 

70 

9337  50 

21 

922  50 

71 

9597  .  50 

22 

1017.50 

72 

9857.50 

23 

w  5 

27  5 

122  5 

1112  50 

73 

10117.50 

24 

1235.00 

74 

w.  13 

27.5 

287.5 

10377.50 

25 

1357  50 

75 

10665.00 

26 

1480  00 

76 

10952.50 

27 

1602  50 

77 

11240.00 

28 

1725.00 

78 

11527.50 

29 
30 





1847.50 
1970  00 

79 

80 

w.  14 

27.5 

315.0 

11815.00 
12130.00 

31 

2092  50 

81 

12445.00 

32 

w  6 

17  5 

1400 

2215  00 

82 

12760  00 

33 

2355  00 

83 

13075  .  00 

34 

2495  00 

84 

13390.00 

35 

2635  .  00 

85 

13705.00 

36 

2775  00 

86 

14020.00 

37 
38 

w.  7 

17.5 

157.5 

2915.00 
3072.50 

87 
88 

w.  15 

ii'.s 

332.5 

14335.00 
14650.00 

39 

3230  00 

89 

14982.50 

40 

3387.50 

90 

15315.00 

41 

3545  00 

91 

15647.50 

42 

3702  .  50 

92 

15980.00 

43 
44 

w.  8 

17.5 

175.0 

3860.00 
4035.00 

93 
94 

w.  16 

17.5 

350.0 

16312.50 
16662.50 

45 
46 

4210.00 
4385  00 

95 
96 

17012.50 
17362.50 

47 

4560.00 

97 

17712.50 

48 
49 
50 

w.  9 

17.5 

192.5 

4735.00 
4927.50 
5120  00 

98 
99 
100 

w'.  17 

17^5 

367.5 

18062.50 
18412.50 
18780.00 

LIVE-LOAD    STRESSES  81 

COMMON  STANDARD  100'-150'  COMMON  STANDARD  150'-200' 


Length 

Wheel 

Load 

Load 
Sums 

Moment 
Sums 

Length 

Load 

Load 
Sums 

Moment 
Sums 

100 

18780.00 

150 

487.5 

40061.25 

101 

19147.50 

151 

490.0 

40550.00 

102 

19515.00 

152 

492.5 

41041.25 

103 

19882.50 

153 

495.0 

41535.00 

104 

W.  18 

17.5 

385^6 

20250.00 

154 

497.5 

42031.25 

105 

20635  .  00 

155 

500.0 

42530.00 

106 

21020.00 

156 

502.5 

43031.25 

107 

21405.00 

157 

505.0 

43535.00 

108 

21790.00 

158 

507.5 

44041.25 

109 

385  '.  6 

22175.00 

159 

510.0 

44550.00 

110 

387.5 

22561.25 

160 

512.5 

45061.25 

111 

390.0 

22950.00 

161 

515.0 

45575.00 

112 

392.5 

23341.25 

162 

517.5 

46091.25 

113 

395.0 

23735.00 

163 

520.0 

46610.00 

114 

397.5 

24131.25 

164 

522.5 

47131.25 

115 

400.0 

24530.00 

165 

525.0 

47655.00 

116 

a 

402.5 

24931.25 

166 

+j 

527.5 

48181.25 

117 

§ 

405.0 

25335.00 

167 

M 

530.0 

48710.00 

118 

W 

407.5 

25741.25 

168 

532.5 

49241.25 

119 

1 

410.0 

26150.00 

169 

0 

& 

VI 

535.0 

49775.00 

120 

•a 

412.5 

26561.25 

170 

1 

537.5 

50311.25 

121 

d 

H 

415.0 

26975.00 

171 

g 

540.0 

50850.00 

122 

1 

417.5 

27391.25 

172 

a 

542.5 

51391.25 

123 

M4 

420.0 

27810.00 

173 

o 

545.0 

51935.00 

124 

o 

422.5 

28231.25 

174 

1C 

547.5 

52481.25 

125 

<>f 

425.0 

28655.00 

175 

of 

550.0 

53030.00 

126 

427.5 

29081.25 

176 

II 

552.5 

53581.25 

127 

430.0 

29510.00 

177 

T3 

555.0 

54135.00 

128 

li 

432.5 

29941.25 

178 

O 

557.5 

54691.25 

129 

nS 

435.0 

30375.00 

179 

g 

560.0 

55250.00 

130 

.  a 

437.5 

30811,25 

180 

£ 

562.5 

55811.25 

131 

1 

440.0 

31250.00 

181 

'3 

565.0 

56375.00 

132 

c 

442.5 

31691.25 

182 

P 

567.5 

56941.25 

133 

P 

445.0 

32135.00 

183 

570.0 

57510.00 

134 

447.5 

32581.25 

184 

572.5 

58081.25 

135 

450.0 

33030.00 

185 

575.0 

58655.00 

136 

452.5 

33481.25 

186 

577.5 

59231.25 

137 

455.0 

33935.00 

187 

580.0 

59810.00 

138 

457.5 

34391.25 

188 

582.5 

60391.25 

139 

460.0 

34850.00 

189 

585.0 

60975.00 

140 

462.5 

35311.25 

190 

587.5 

61561.25 

141 

465.0 

35775.00 

191 

590.0 

62150.00 

142 

467.5 

36241.25 

192 

592.5 

62741.25 

143 

470.0 

36710.00 

193 

595.0 

63335.00 

144 

472.5 

37181.25 

194 

597.5 

63931.25 

145 

475.0 

37655.00 

195 

600.0 

64530.00 

146 

477.5 

38131.25 

196 

602.5 

65131.25 

147 

480.0 

38610.00 

197 

605.0 

65735.00 

148 

482.5 

39091.25 

198 

607.5 

66341.25 

149 

485.0 

39575.00 

199 

610.0 

66950.00 

150 

487.5 

40061.25 

200 

612.5 

67561.25 

82 


LIVE-LOAD    STRESSES 


COMMON  STANDARD  200 '-250' 


COMMON  STANDARD  250 '-300' 


Length 

Load 

Load 
Sums 

Moment 
Sums 

Length 

Load 

Load 
Sums 

Moment 
Sums 

200 

612.5 

67561.25 

250 

737.5 

101311.25 

201 

615.0 

68175.00 

251 

740.0 

102050.00 

202 

617.5 

68791.25 

252 

742.5 

102791.25 

203 

620.0 

69410.00 

253 

745.0 

103535.00 

204 

622.5 

70031.25 

254 

747.5 

104281.25 

205 

625.0 

70655.00 

255 

750.0 

105030.00 

206 

627.5 

71281.25 

256 

752.5 

105781.25 

207 

630.0 

71910.00 

257 

755.0 

106535.00 

208 

632.5 

72541.25 

258 

757.5 

107291.25 

209 

635.0 

73175.00 

259 

760.0 

108050.00 

210 

637.5 

73811.25 

260 

762.5 

108811.25 

211 

640.0 

74450.00 

261 

765.0 

109575.00 

212 

642.5 

75091.25 

262 

767.5 

110341.25 

213 

645.0 

75735.00 

263 

770.0 

111110.00 

214 

647.5 

76381.25 

264 

772.5 

111881.25 

215 

650.0 

77030.00 

265 

775  '.0 

112655.00 

216 

+3 

652.5 

77681.25 

266 

•g 

777.5 

113431.25 

217 

1 

655.0 

78335.00 

267 

1 

780.0 

114210.00 

218 

(_, 

657.5 

78991.25 

268 

(H 

782.5 

114991.25 

219 

& 

660.0 

79650.00 

269 

I 

785.0 

115775.00 

220 

•a 

662.5 

80311.25 

270 

•3 

787.5 

116561.25 

221 

£j 

665.0 

80975.00 

271 

§ 

790.0 

117350.00 

222 

8< 

667.5 

81641.25 

272 

a 

792.5 

118141.25 

223 

Q 

670.0 

82310.00 

273 

795.0 

118935.00 

224 

io 

672.5 

82981.25 

274 

g 

797.5 

119731.25 

225 

of 

675.0 

83655.00 

275 

c<f 

800.0 

120530.00 

226 

II 

677.5 

84331.25 

276 

II 

802.5 

121331.25 

227 

1 

680.0 

85010.00 

277 

1i 

805.0 

122135.00 

228 

J 

682.5 

85691.25 

278 

o 

807.5 

122941.25 

229 

hH 

685.0 

86375.00 

279 

" 

810.0 

123750.00 

g 

3 

230 

I 

687.5 

87061.25 

280 

1 

812.5 

124561.25 

231 

•g 

690.0 

87750.00 

281 

"c 

815.0 

125375.00 

232 

p 

692.5 

88441.25 

282 

P 

817.5 

126191.25 

233 

695.0 

89135.00 

283 

820.0 

127010.00 

234 

697.5 

89831.25 

284 

822.5 

127831.25 

235 

700.0 

90530.00 

285 

825.0 

128655.00 

236 

702.5 

91231.25 

286 

827.5 

129481.25 

237 

705.0 

91935.00 

287 

830.0 

130310.00 

238 

707.5 

92641.25 

288 

832.5 

131141.25 

239 

710.0 

93350.00 

289 

835.0 

131975.00 

240 

712.5 

94061.25 

290 

837.5 

132811.25 

241 

715.0 

94775.00 

291 

840.0 

133650.00 

242 

717.5 

95491.25 

292 

842.5 

134491.25 

243 

720.0 

96210.00 

293 

845.0 

135335.00 

244 

722.5 

96931.25 

294 

847.5 

136181.25 

245 

725.0 

97655.00 

295 

850.0 

137030.00 

246 

727.5 

98381.25 

296 

852.5 

137881.25 

247 

730.0 

99110.00 

297 

855.0 

138735.00 

248 

732.5 

99841.25 

298 

857.5 

139591.25 

249 

735.0 

100575.00 

299 

860.0 

140450.00 

250 

737.5 

101311.25 

300 

862.5 

141311.25 

LIVE-LOAD    STRESSES  83 

COMMON  STANDARD  300'-350'  COMMON  STANDARD  350'-400' 


Length 

Load 

Load 
Sums 

Moment 
Sums 

Length 

Load 

Load 
Sums 

Moment 
Sums 

300 

862.5 

141311.25 

350 

987.50 

187561.25 

301 

865.0 

142175.00 

351 

990.00 

188550.00 

302 

867.5 

143041.25 

352 

992.50 

189541.25 

303 

870.0 

143910.00 

353 

995.00 

190535.00 

304 

872.5 

144781.25 

354 

997.50 

191531.25 

305 

875.0 

145655.00 

355 

1000.00 

192530.00 

306 

877.5 

146531.25 

356 

1002.50 

193531.25 

307 

880.0 

147410.00 

357 

1005.00 

194535.00 

308 

882.5 

148291.25 

358 

1007.50 

195541.25 

309 

885.0 

149175.00 

359 

1010.00 

196550.00 

310 

887.5 

150061.25 

360 

1012.50 

197561.25 

311 

890.0 

150950.00 

361 

1015.00 

198575.00 

312 

892.5 

151841.25 

362 

1017.50 

199591.25 

313 

895.0 

152735.00 

363 

1020.00 

200610.00 

314 

897.5 

153631.25 

364 

1022.50 

201631.25 

315 

900.0 

154530.00 

365 

1025.00 

202655.00 

316 

902.5 

155431.25 

366 

1027.50 

203681.25 

317 

4d 

905.0 

156335.00 

367 

4> 

1030.00 

204710.00 

318 

8 

907.5 

157241.25 

368 

§ 

1032.50 

205741.25 

319 

•M 

Ld 

910.0 

158150.00 

369 

IM 

£ 

1035.00 

206775.00 

320 

ft 

912.5 

159061.25 

370 

£ 

1037.50 

207811.25 

321 

915.0 

159975.00 

371 

-8 

1040.00 

208850.00 

322 

917.5 

160891.25 

372 

s 

1042.50 

209891.25 

323 

8, 

920.0 

161810.00 

373 

i 

1045.00 

210935.00 

324 

A 

922.5 

162731.25 

374 

M< 

1047.50 

211981.25 

325 

S 

925.0 

163655.00 

375 

o 

1050.00 

213030.00 

326 

~ 

927.5 

164581.25 

376 

^ 

1052  .  50 

214081.25 

327 

0s! 

930.0 

165510.00 

377 

c^ 

1055.00 

215135.00 

328 

-d 

932.5 

166441.25 

378 

T3 

1057.50 

216191.25 

329 

0 

985.0 

167375.00 

379 

J 

1060.00 

217250.00 

330 

a 

937.5 

168311.25 

380 

Ej 

1062.50 

218311.25 

331 

E 

o 

940.0 

169250.00 

,381 

§ 

1065.00 

219375.00 

332 

1 

942.5 

170191.25 

382 

1 

1067.50 

220441.25 

333 

1 

945.0 

171135.00 

383 

p 

;5 

1070.00 

221510.00 

334 

947.5 

172081.25 

384 

1072.50 

222581.25 

335 

950.0 

173030.00 

385 

1075  .  00 

223655.00 

336 

952.5 

173981.25 

386 

1077.50 

224731.25 

337 

955.0 

174935.00 

387 

1080.00 

225810.00 

338 

957.5 

175891.25 

388 

1082.50 

226891.25 

339 

960.0 

176850.00 

389 

1085.00 

227975.00 

340 

962.5 

177811.25 

390 

1087.50 

229061.25 

341 

965.0 

178775.00 

391 

1090.00 

230150.00 

342 

967.5 

179741.25 

392 

1092.50 

231241.25 

343 

970.0 

180710.00 

393 

1095.00 

232335.00 

344 

972.5 

181681.25 

394 

1097.50 

233431.25 

345 

975.0 

182655.00 

395 

1100.00 

234530.00 

346 

977.5 

183631.25 

396 

1102.50 

235631.25 

347 

980.0 

184610.00 

397 

1105.00 

236735.00 

34S 

982.5 

185591  .25 

398 

1107.50 

237841.25 

349 

985.0 

186575.00 

399 

1110.00 

238950.00 

350 

987.5 

187561.25 

400 

1112.50 

240061.25 

84  LIVE-LOAD    STRESSES 

LACKAWANNA  0'-50'  LACKAWANNA  50'-100' 


Length 

Wheel 

Load 

Load 
Sums 

Moment 
Sums 

Length 

Wheel 

Load 

Load 

Sums 

Moment 
Sums 

0 
1 

W.  1 

11 

11.00 

00.000 
11  000 

50 

51 

4744.000 
4911  000 

2 

22.000 

52 

5078  000 

3 

33  000 

53 

5245  000 

4 

44  000 

54 

w  10 

11 

178  00 

5412  000 

5 

55  000 

55 

5590  000 

6 

7 

W.  2 

25 

36.66 

66.000 
77  000 

56 
57 

5768.000 
5946  000 

8 

113.000 

58 

6124  000 

9 
10 

149.000 
185.000 

59 
60 

6302.000 
6480  000 

11 
12 
13 

'  W.'  3 

25 

'ei'.oo 

221.000 
257.000 
318  000 

61 
62 
63 

W.  11 

25 

203.00 

6658.000 
6861.000 
7064  000 

14 

379  000 

64 

7267  000 

15 

440  000 

65 

7470  000 

16 
17 

w.  4 

25 

86.66 

501.000 
562.000 

66 
67 

w.  12 

25 

228.66 

7673.000 
7901  000 

18 

648.000 

68 

8129  000 

19 

734.000 

69 

8357  000 

20 

820.000 

70 

8585  000 

21 
22 
23 

'  w.'  5' 

25 

iii'.oo 

906.000 
992.000 
1103  000 

71 
72 
73 

w.  13 

25 

253.00 

8813.000 
9066.000 
9319  000 

24 
25 



1214.000 
1325  000 

74 
75 

9572.000 
9825  000 

26 
37 

1436.000 
1547.000 

76 

77 

w.  14 

25 

278.00 

10078.000 
10356  000 

28 

1658.000 

78 

10634  000 

29 

1769  000 

79 

10912  000 

30 

1880.000 

80 

11190  000 

31 
32 

w.  6 

14 

125.00 

1991.000 
2116  000 

81 

82 

11468.000 
11746  000 

33 

2241  000 

83 

12024  000 

34 

2366  000 

84 

12302  000 

35 
36 
37 

w.  7 

i4 

139'.00 

2491.000 
2616.000 
2755  .  000 

85 
86 
87 

w.  15 

14 

292.00 

12580.000 
12872.000 
13146  000 

38 
39 

2894.000 
3033.000 

88 
89 

13456.000 
13748  000 

40 
41 
42 
43 
44 

'  w.'  8' 

14 

153.00 

3172.000 
3311.000 
3464.000 
3617.000 
3770  000 

90 
91 
92 
93 
94 

w.  16 

14 

306.00 

14040.000 
14346.000 
14652.000 
14958.000 
15264  000 

45 

3923  000 

95 

w  17 

14 

320.00 

15570  000 

46 

47 

w.  9 

14 

167.00 

4076.000 
4243.000 

96 
97 

15890.000 
16210  000 

48 

4410  000 

98 

16530  000 

49 

4577  000 

99 

16850  000 

50 

4744.000 

100 

w.  18 

14 

334.00 

17170.000 

LIVE-LOAD    STRESSES  85 

LACKAWANNA  100'-150'  LACKAWANNA  150'-200' 


Length 

Wheel 

Load 

Load 

Sums 

Moment 
Sums 

Length 

Load 

Load 

Sums 

Moment 
Sums 

103 

W.  18 

14 

334.00 

17170.000 

150 

437.50 

36250.500 

101 

17504.000 

151 

439.75 

36689.125 

102 

17838.000 

152 

442.00 

37130.000 

103 

18172  .  000 

153 

444.25 

37573.125 

104 

334^00 

18506.000 

154 

446.50 

38018.500 

105 

336.25 

18841  .  125 

155 

448.75 

38466.125 

106 

338.50 

19178.500 

156 

451.00 

38916.000 

107 

340.75 

19518.125 

157 

453.25 

39368.125 

108 

343.00 

19860.000 

158 

455.50 

39822.500 

109 

345.25 

20204.125 

159 

457.75 

40279.125 

110 

347.50 

20550.500 

160 

460.00 

40738.000 

111 

349.75 

20899.125 

161 

462.25 

41199.125 

112 

352.00 

21250.000 

162 

464.50 

41662.500 

113 

354.25 

21603  .  125 

163 

466.75 

42128.125 

114 

356.50 

21958.500 

164 

469.00 

42596.000 

115 

358.75 

22316.125 

165 

471.25 

43066.125 

116 

361.00 

22676.000 

166 

473.50 

43538.500 

117 

t 

363.25 

23038.125 

167 

43 
0 

475.75 

44013  .  125 

118 

a 

365.50 

23402.500 

168 

«2 

478.00 

44490.000 

119 

& 

367.75 

23769.125 

169 

I 

480.25 

44969  .  125 

120 

02 

370.00 

24138.000 

170 

CQ 

482.50 

45450.500 

121 

T3 

372.25 

24509.125 

171 

TS 

a 

484.75 

45934.125 

122 

§ 

374.50 

24882.500 

172 

487.00 

46420.000 

123 

o 
R 

376.75 

25258.125 

173 

a 

489.25 

46908.125 

124 

o 

379.00 

25636.000 

174 

o 

491.50 

47398.500 

125 

<N 

381.25 

26016.125 

175 

c3. 

493.75 

47891  .  125 

126 

c^f 

383.50 

26398.500 

176 

c<f 

496.00 

48386.000 

127 

II 

385.75 

26783.125 

177 

II 

498.25 

48883  .  125 

128 

T3 

388.00 

27170.000 

178 

73 

500.50 

49382.500 

129 

g 

3 

390.25 

27559  .  125 

179 

1 

502.75 

49884.125 

130 

392.50 

27950.500 

180 

J- 

505.00 

50338.000 

131 

a 

394.75 

28344.125 

181 

B 

507.25 

50894.125 

132 

1 

397.00 

28740.000 

182 

£ 

509.50 

51402.500 

133 

'A 

399.25 

29138.125 

183 

1 

511.75 

51913.125 

134 

M 

401.50 

29538.500 

184 

M* 

514.00 

52426.000 

135 

403.75 

29941  .  125 

185 

516.25 

52941  .  125 

136 

406.00 

30346.000 

186 

518.50 

53458.500 

137 

408.25 

30753  .  125 

187 

520.75 

53978.125 

138 

410.50 

31162.500 

188 

523.00 

54500.000 

139 

412.75 

31574.125 

189 

525.25 

55024.125 

140 

415.00 

31988.000 

190 

527.50 

55550.500 

141 

417.25 

32404.125 

191 

529.75 

56079  .  125 

142 

419.50 

32882.500 

192 

532.00 

56610.000 

143 

421.75 

33243  .  125 

193 

534.25 

57143.125 

144 

424.00 

33666.000 

194 

536.50 

57678.500 

145 

426.25 

34091  .  125 

195 

538.75 

58216.125 

146 

428.50 

34518.500 

196 

541.00 

58756.000 

147 

430.75 

34948.125 

197 

543.25 

59298.125 

148 

433.00 

35380.000 

198 

545.50 

59842.500 

149 

435.25 

35814.125 

199 

547.75 

60389.125 

150 

437.50 

36250.500 

200 

550.00 

60938.000 

86  LIVE-LOAD    STRESSES 

LACKAWANNA  200'-250'  LACKAWANNA  250'-300' 


Length 

Load 

Load 
Sums 

Moment 
Sums 

Length 

Load 

Load 
Sums 

Moment 
Sums 

200 

550.00 

60938.000 

250 

662.50 

91250.500 

201 

552.25 

61489.125 

251 

664.75 

91914.125 

202 

554.50 

62042.500 

252 

667.00 

92580.000 

203 

556.75 

62598.125 

253 

669.25 

93248.125 

204 

559.00 

63156.000 

254 

671.50 

93918.500 

205 

561.25 

63716.125 

255 

673.75 

94591  .  125 

206 

563.50 

64278.500 

256 

676.00 

95266.000 

207 

565.75 

64843.125 

257 

678.25 

95943  .  125 

208 

568.00 

65410.000 

258 

680.50 

96622.500 

209 

570.25 

65979.125 

259 

682.75 

97304.125 

210 

572.50 

66550.500 

260 

685.00 

97988.000 

211 

574.75 

67124.125 

261 

687.25 

98674.125 

212 

577.00 

67700.000 

262 

689.50 

99362.500 

213 

579.25 

68278.125 

263 

691.75 

100053  .  125 

214 

581,50 

68858.500 

264 

694.00 

100746.000 

215 

583.75 

69441  .  125 

265 

696.25 

101441.125 

216 

586.00 

70026.000 

266 

698.50 

102138.500 

217 

4g 

588.25 

70613.125 

267 

jg 

700.75 

102838  .  125 

218 

o 
•  £ 

590.50 

71202.500 

268 

J 

703.00 

103540.000 

219 

<•»—  ( 

Lg 

592.75 

71794.125 

269 

;., 

705.25 

104244.125 

220 

I 

rf) 

595.00 

72388.000 

270 

& 

02 

707.50 

105950.500 

221 

fi 

597.25 

72984.125 

271 

T3 

r* 

709.75 

105659  .  125 

222 

;3 

599.50 

73582.500 

272 

712.00 

106370.000 

223 

a 

601.75 

74183.125 

273 

ft 

714.25 

107083.125 

224 

o 

604.00 

74786.000 

274 

S 

716.50 

107798.500 

225 

% 

606.25 

75391  .  125 

275 

IQ 

<N 

718.75 

108516.125 

226 

c$ 

608.50 

75998.500 

276 

IN" 

721.00 

109236.000 

227 

II 

610.75 

76608.125 

277 

II 

723.25 

109958.125 

228 

T3 

613.00 

77220.000 

278 

T3 

725.50 

110682.500 

229 

I 

615.25 

77834.125 

279 

J 

727.75 

111409.125 

230 

g 

617.50 

78450.500 

280 

g 

730.00 

112138.000 

231 

C 

Q 

619.75 

79069.125 

281 

o 

732.25 

112869.125 

232 

622.00 

79690.000 

282 

i 

734.50 

113602.500 

233 

'S 
£3 

624.25 

80313.125 

283 

a 

J3 

736.75 

114338.125 

234 

626.50 

80938.500 

284 

739.00 

115076.000 

235 

628.75 

81566.125 

285 

741.25 

115816.125 

236 

631.00 

82196.000 

286 

743.50 

116558.500 

237 

633.25 

82828.125 

287 

745.75 

117303.125 

238 

635.50 

83462.500 

288 

748.00 

118050.000 

239 

637.75 

84099.125 

289 

750.25 

118799.125 

240 

640.00 

84738.000 

290 

752.50 

119550.500 

241 

642.25 

85379.125 

291 

754.75 

120304.125 

242 

644.50 

86022.500 

292 

757.00 

121060.000 

243 

646.75 

86668.125 

293 

759.25 

121818.125 

244 

649.00 

87316.000 

294 

761.50 

122578.500 

245 

651.25 

87966.125 

295 

763.75 

123341  .  125 

246 

653.50 

88618.500 

296 

766.00 

124106.000 

247 

655.75 

89273.125 

297 

768.25 

124873  .  125 

248 

658.00 

89930.000 

298 

770.50 

125642.500 

249 

660.25 

90589.125 

299 

772.75 

126414.125 

250 

662.50 

91250.500 

300 

775.00 

127188.000 

1 

LIVE-LOAD    STRESSES  87 

LACKAWANXA  300'-350'  LACKAWANNA  350'-400' 


Length 

T-_J     Load 
Load    Sums 

Moment 
Sums 

Length 

Load 

Load 

Sums 

Moment 
Sums 

300 

I  - 

775.00 

127188.000 

350 

887.50 

168750.500 

301 

777.25 

127964.125 

351 

889.75 

169639  .  125 

302 

779.50 

128742.500 

352 

892.00 

170530.000 

303 

781.75 

129523.125 

353 

894.25 

171423  .  125 

304 

784.00 

130306.000 

354 

896.50 

172318.500 

305 

786.25 

131091.125 

355 

898.75 

173216.125 

306 

788.50 

131878.500 

356 

901.00 

174116.000 

307 

790.75 

132668.125 

357 

903.25 

175018.125 

308 

793.00 

133460.000 

358 

905.50 

175922.500 

309 

795.25 

134254.125 

359 

907.75 

176829.125 

310 

797.50 

135050.500 

360 

910.00 

177738.000 

311 

799.75 

135849.125 

361 

912.25 

178649.125 

312 

802.00 

136650.000 

362 

914.50 

179562.500 

313 

804.25 

137453  .  125 

363 

916.75 

180478.125 

314 

806.50 

138258.500 

364 

919.00 

181396.000 

315 

808.75 

139066  .  125 

365 

921.25 

182316.125 

316 

811.00 

139876.000 

366 

923.50 

183238.500 

317 

*• 

813.25 

140688.125 

367 

4J 

925.75 

184163.125 

318 

8 

815.50 

141502.500 

368 

8 

928.00 

185090.000 

319 

«*H 

817.75 

142319.125 

369 

C+H 

*, 

930.25 

186019.125 

O 

I. 

320 

rr. 

820.00 

143138.000 

370 

&C 

t/5 

932.50 

186950.500 

321 

T3 

822.25 

143959.125 

371 

43 

934.75 

187884.125 

322 

fl 

3 

824.50 

144782.500 

372 

d 

937.00 

188820.000 

323 

& 

826.75 

145608.125 

373 

a, 

939.25 

189758.125 

324 

Q 

829.00 

146436.000 

374 

Q 

941.50 

190698.500 

325 

H 

831.25 

147266.125 

375 

of 

943.75 

191641.125 

326 

C^f 

833.50 

148098.500 

376 

fff 

946.00 

192586.000 

327 

II 

835.75 

148933.125 

377 

II 

948.25 

193533.125 

328 

838.00 

149770.000 

378 

nM 

950.50 

194482.500 

329 

1 

840.25 

150609.125 

379 

0 

952.75 

195434.125 

330 

842.50 

151450.500 

380 

g 

955.00 

196388.000 

331 

M 

844.75 

152294.125 

381 

957.25 

197344.125 

332 

.§ 

847.00 

153140.000 

382 

£ 

959.50 

198302.500 

333 

« 

h-^ 

849.25 

153988.125 

383 

'3 
p 

961.75 

199263  .  125 

334 

t—  ' 

851.50 

154838.500 

384 

964.00 

200226.000 

335 

853.75 

155691  .  125 

385 

966.25 

201191.125 

336 

. 

856.00 

156546.000 

386 

968.50 

202158.500 

337 

858.25 

157403  .  125 

387 

970.75 

203128.125 

338 

860.50 

158262.500 

388 

973.00 

204100.000 

339 

882.75 

159124.125 

389 

975.25 

205074.125 

340 

865.00 

159988.000 

390 

977.50 

206050.500 

341 

867.25 

160854  .  125 

391 

979.75 

207029.125 

342 

869.50 

161722.500 

392 

982.00 

208010.000 

343 

871.75 

162593  .  125 

393 

984.25 

208993.125 

344 

874.00 

183466.000 

394 

986.50 

209978.500 

345 

876.25 

164341  .  125 

395 

988.75 

210966.125 

346 

878.50 

165218.500 

396 

991.00 

211956.000 

347 

880.75 

166098.125 

397 

993.25 

212948.125 

348 

883.00 

166980.000 

398 

995  .  50 

213942.500 

349 

885.25 

167864.125 

399 

997.75 

214939.125 

350 

887.50 

168750.500 

400 

1000.00 

215938.000 

88 


LIVE-LOAD    STRESSES 


TABLE  3 

POSITION  OF  COOPER'S  LOADINGS  FOR  MAXIMUM  STRESS 
Shorter  Segment  h 


Segments 

1C 

0 

3 

§ 

a 

§ 

§ 

0 

5 

S 

3 

8 

S 

- 

1O 

i 

s 

§ 

to 
« 

§ 

0 

§ 

1 

o 

TT 

30C 
25C 
19C 

49 

a 

<D 

1 

h-260 
1-200 
f-150 
140 
130 
120 
110 
100 
95 
90 
85 
80 
75 
70 
65 
60 
55 
50 
45 
40 
35 
30 
25 
20 
15 
10 
5 

2 
2 
2 

11 
11 
11 

2 
2 

2 

2 
2 

3 
3 
3 
3 
3 
3 
3 
12 

3 
3 
3 
3 
3 
4 
3 
3 

3 
3 
3 
3 
3 
3 

12 
12 
3 
3 
3 
3 
3 
3 

4 
4 
4 
4 

4 
4 
4 
4 

12 
12 

3 

4 
4 

4 
4 

4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 

4 

4 
4 
4 

12 
12 

4 
4 
4 

4 
4 
4 
4 
4 
4 
4 

e 

•  j 

B 

5 

13 

13 
13 
13 
13 

* 

5 

E 
E 

B 
e 

(i 

6 

13 
13 
13 
13 

6 
6 

e 

(5 
6 
ft 
7 

li 
13 
13 
12 
12 

7 
7 

7 
7 
7 

7 

7 
It 
13 

13 
13 
12 

4 

8 

8 
8 
8 
8 

14 
13 
13 
13 
1  2 

8 
8 

9 
9 
9 

9 

13 

13 

i3 

9 

9 

9 

10 
10 
10 
10 
13 
13 
13 
13 
12 

K 

11 
11 
11 
11 
11 
11 
13 

id 

13 
12 

K 
11 
11 
12 
12 
12 
12 
12 
12 
12 

Y2 
12 

n 
n 

i 
i 

i 
i 

11 

12 
12 
12 
12 
12 
12 
12 
12 
12 
12 

12 
12 
12 
12 
12 
13 
13 
13 
13 
13 

12 
12 
12 
12 
12 
13 
13 
13 
13 

13 

13 
13 
1 

1 
1 

1 
1 

14 
14 
14 
14 

14 
14 
14 

E 

r 
O 

Fj 
Fj 

e 
0 

r 

»J 

17 
17 
17 

17 
17 

18 
IS 
18 
18 

... 

13 

13 
13 

12 
12 

12 
12 

12 
12 

12 
12 

12 

11 

12 
11 
11 

12 

11 

12 
12 
12 
12 
13 
13 

13 
13 
13 
13 
13 

Y2 
T3 
13 

12 
lY 

Y3 
13 

fl 
13 
12 

11 
11 
11 

11 
11 

GENERAL  NOTES. — The  table  gives  wheel  for  maximum  for  any  stress  which  has  a  triangular 
influence  line. 

In  case  of  two  unequal  segments,  the  live  load  approaches  on  the  longer  segment  except 
where  wheel  is  overlined,  when  live  load  approaches  on  shorter  segment. 

When  both  segments  are  each  greater  than  142  ft.,  advance  load  on  longer  segment  first, 
and  upon  next  segment  until  wheel  No.  1  is  within  33  feet  of  the  far  end  of  the  latter. 


LIVE-LOAD    STRESSES 


89 


TABLE  4 

POSITION  OF  COOPER'S  LOADINGS  FOR  ABSOLUTE  MAXIMUM  BENDING  MOMENT 
IN  GIRDER  BRIDGES  WITHOUT  PANELS 

5  —  Span  in  feet. 

c  =  Distance  in  feet  that  wheel  No.  1  has  moved  to  left  beyond  centre  of 
span. 

w  =  wheel  under  which  absolute  maximum  bending  moment  occurs. 
a  =  distance  that  w  is  to  left  from  centre  of  span. 

6  =       "  "    w     "     right  "          "      "      " 


S 

c 

w 

a 

b 

0'  to  8'.5 

S'.OO 

2 

O'.OO 

8.5   11.1 

9.25 

2 

1.25 

11.1   18.7 

13.00 

3 

0.00 

18.7   27.6 

14.25 

3 

1.25 

27.6   34.9 

13.39 

3 

0.39 

34.9   38.7 

17.06 

4 

, 

0.94 

38.7   48.6 

18.21 

4 

0.2! 

.. 

48.6   53.7 

19.45 

4 

1.45 

>t 

53.7   58.4 

74.13 

13 

0.13 

58.4   63.2 

75.37 

13 

1.37 

63.2   70.00 

74.07 

13 

0.07 

.... 

NOTE. — For  spans  greater  than  70  feet,  the  maximum  centre  moment  equals  the  absolute 
maximum  bending  moment  with  an  error  of  less  than  one  per  cent. 


TABLE  5 

POSITION  OF  COOPER'S  LOADINGS  FOR  MAXIMUM  END  SHEAR  IN  GIRDER 
BRIDGES  WITHOUT  PANELS 


Span 

Direction  Load 
Moves 

Position  of 
Load 

Location  of 
Maximum  Shear 

O'to    23' 
23    "     27 
27    "    46 
46    "     62 
62    "  400 

Right  to  left 
Right  to  left 
Right  to  left 
Right  to  left 
Right  to  left 

wz   at  left  end 
Wn   at  right  end 
Wz  at  left  end 
w\\  at  left  end 
w-z   at  left  end 

Left  end 
Right  end 
Left  end 
Left  end 
Left  end 

90 


LIVE-LOAD    STRESSES 


TABLE  6 

POSITION  OF  COOPER'S  LOADINGS  FOR  MAXIMUM  SHEAR  IN  PANELS  OF  GIRDER 
AND  TRUSS  BRIDGES 


Number  of 
Panels 

Panel 

PANEL  LENGTH  IN  FEET                    • 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

6  

7 

0-1 
1-2 
2-3 
3-4 
4-5 
0-1 
1-2 
2-3 
3^ 
4-5 
5-6 
0-1 
1-2 
2-3 
3-4 
4-5 
5-6 
6-7 
0-1 
1-2 
2-3 
3^t 
4-5 
5-6 
6-7 
7-8 
0-1 
1-2 
2-3 
3¥4 
4-5 
5-6 
6-7 
7-8 
8-9 

4 
3 
3 
2 
2 
4 
3 
3 
3 
2 
2 
3 
3 
3 
3 
2 
2 
2 
3 
3 
3 
3 
2 
2 
2 
2 
3 
3 
3 
3 
3 
2 
2 
2 
1 

4 
3 
3 
2 
2 
4 
3 
3 
3 
2 
2 
4 
3 
3 
3 
2 
2 
2 

5 
4 
3 
3 
2 
4 
4 
3 
3 
2 
2 

5 
4 
3 
3 
2 
5 
4 
3 
3 
2 

2 
5 
4 
4 
3 
3 
2 
2 

5 
4 
3 
3 
2 
5 
4 
4 
3 
3 
2 
5 
4 
4 
3 
3 
2 
2 
5 

5 
4 
4 
3 
2 
5 
4 
4 
3 
3 
2 
5 
4 
4 
3 
3 
2 
2 
5 

3 
3 
2 
2 
4 
3 
3 
3 
2 
2 
4 
3 
3 
3 
2 
2 

2 

3 
3 
2 
2 
4 
3 
3 
3 
2 
2 
4 
3 
3 
3 
2 
2 
2 

4 
3 
2 
2 
4 
4 
3 
3 
2 
2 
4 
4 
3 
3 
2 
2 
2 

4 
3 
2 
2 
4 
4 
3 
3 
2 
2 
4 
4 
3 
3 
3 
2 
2 

4 
3 
2 
2 
4 
4 
.3 
3 
2 
2 

4 
3 
2 
2 
4 
4 
3 
3 
2 
2 

4 
3 
2 
2 
4 
4 
3 
3 
2 
2 

4 
3 
3 
2 
4 
4 
3 
3 
2 
2 

8  
9  

10.  ..    . 

4 
3 
3 
3 
2 
2 

4 
3 
3 
3 
2 
2 

4 
3 
3 
3 
2 
2 

4 
4 
3 
3 
2 
2 

4 
4 
3 
3 
2 
2 

3 
3 
3 
3 
2 
2 
2 
4 
3 
3 
3 
3 
2 
2 
2 
1 

3 
3 
3 
3 
2 
2 
2 
4 
3 
3 
3 
3 
2 
2 
2 
1 

3 
3 
3 
3 
2 
2 
2 
4 
3 
3 
3 
3 
2 
2 
2 
1 

4 
3 
3 
3 
2 
2 
2 
4 
4 
3 
3 
3 
2 
2 
2 
1 

4 
3 
3 
3 
2 
2 
2 
4 
4 
3 
3 
3 
2 
2 
2 
1 

4 
3 
3 
3 
2 
2 
2 
4 
4 
3 
3 
3 
3 
2 
2 
1 

3 
3 
3 
2 
2 
2 
4 
4 
3 
3 
3 
3 
2 
2 
1 

4 
3 
3 
2 
2 
2 
4 
4 
4 
3 
3 
3 
2 
2 
2 

4 
3 
3 
3 
2 
2 
4 
4 
4 
3 
3 
3 
2 
2 
2 

4 
3 
3 
3 
2 
2 
4 
4 
4 
3 
3 
3 
2 
2 

2 

4 
3 
3 
3 
2 
2 
4 
4 
4 
3 
3 
3 
2 

2 

2 

4 
3 
3 
3 
2 
2 
5 
4 
4 
3 
3 
3 
2 
2 
2 

4 
3 
3 
3 
2 
2 
5 
4 
4 
4 
3 
3 
2 
2 
2 

NOTE. — Place  tabulated  wheel  at  right  end  of  corresponding  panel  with  locomotive   ad- 
vancing toward  left. 


LIVE-LOAD    STRESSES  91 

TABLE  7 

MAXIMUM  MOMENTS,   SHEARS,   AND  PIER  REACTIONS  FOR  COOPER'S 
STANDARD  LOADINGS 

(Figures  for  One  Rail) 


Span 

£40 

EoO 

Max. 
Moment 

Max.  Shears 

Max. 
Pier 
React. 

Max. 
Moment 

Max.  Shears 

Max. 
Pier 
React. 

End 

y*  pt. 

Cent. 

End 

MPt. 

Cent. 

10 

56.3 
65.7 
80.0 
95.0 
110.0 
125.0 
140.0 
155.0 
170.0 
186.6 
206.3 
226.0 
245.7 
265.4 
2S5.2 
305.0 
324.8 
344.6 
365.5 
388.0 
410.5 
432.9 
455.4 
477.9 
500.6 
523.0 
548.6 
574.3 
600.0 
625.6 
655.6 
684.6 
713.6 
742.6 
771.6 
800.6 
829.8 
858.6 
887.6 
918.8 
950.9 
983.1 
1015.2 
1047.4 

30.0 
32.7 
35.0 
36.9 
38.6 
40.0 
42.5 
44.7 
46.7 
48.4 
50.0 
51.4 
52.7 
53.9 
55.4 
56.8 
58.1 
59.2 
60.4 
61.6 
63.0 
64.4 
65.7 
66.9 
68.1 
69.2 
70.6 
71.9 
73.1 
74.3 
75.4 
76.8 
78.4 
79.4 
80.6 
81.7 
82.8 
83.8 
85.0 
86.1 
87.2 
88.4 
89.3 
90.5 

20.0 
20.9 
21.7 
22.3 
23.6 
25.0 
26.3 
27.4 
28.3 
29.2 
30.0 
31.4 
32.7 
33.9 
35.0 
38.0 
38.9 
37.8 
38.6 
39.3 
40.0 
40.7 
41.3 
42.0 
42.8 
43.5 
44.1 
44.8 
45.4 
46.0 
48.8 
47.5 
48.2 
48.9 
49.5 
50.1 
50.7 
51.4 
52.1 
52.8 
53.5 
54.1 
54.8 
55.4 

10.0 
10.9 
11.7 
12.3 
12.9 
13.3 
13.7 
13.8 
13.9 
14.0 
14.0 
14.5 
15.0 
15.4 
15.8 
16.2 
16.5 
16.9 
17.1 
17.4 
17.7 
18.2 
18.8 
19.2 
19.7 
20.1 
20.6 
21.0 
21.3 
21.7 
22.0 
22.3 
22.6 
22.9 
23.2 
23.4 
23.7 
23.9 
24.2 
24.5 
24.9 
25.2 
25.5 
25.8 

40.0 
43.7 

46.7 
49.2 
52.2 
54.7 
56.9 
58.8 
60.7 
62.9 
65.6 
68.0 
70.2 
72.2 
74.0 
75.7 
77.7 
80.2 
82.3 
84.4 
86.3 
88.5 
91.0 
93.3 
95.5 
97.5 
99.6 
101.5 
103.7 
105.9 
108.0 
110.0 
112.1 
114.3 
116.5 
118.6 
120.7 
122.7 
124.8 
126.8 
128.7 
131.0 
133.3 
135.6 

70.4 

82.1 
100.0 
118.8 
137.5 
156.3 
175.0 
193.8 
212.5 
233.3 
257.9 
282.5 
307.1 
331.8 
356.5 
381.3 
406.0 
430.8 
456.9 
485.0 
513.0 
541.1 
569.3 
597.4 
625.8 
653.8 
685.8 
717.9 
750.0 
783.3 
819.5 
855.8 
892.0 
928.3 
964.5 
1000.8 
1037.3 
1073.3 
1109.5 
1148.5 
1188.6 
1228.9 
1269.0 
1309.2 

37.5 
40.9 
43.8 
46.2 
48.2 
50.0 
53.1 
55.9 
58.3 
60.5 
62.5 
64.3 
65.9 
67.4 
69.3 
71.0 
72.6 
74.0 
75.5 
76.9 
78.8 
80.5 
82.1 
83.7 
85.1 
86.5 
88.2 
89.8 
91.4 
92.9 
94.3 
96.0 
97.6 
99.2 
100.7 
102.1 
103.5 
104.9 
106.3 
107.7 
109.0 
110.4 
111.8 
113.1 

25.0 
26.1 
27.1 
27.9 
29.5 
31.3 
32.9 
34.3 
35.4 
36.5 
37.5 
39.2 
40.9 
42.4 
43.8 
45.0 
46.1 
47.2 
48.2 
49.1 
50.0 
50.9 
51.8 
52.5 
53.5 
54.4 
55.1 
56.0 
56.7 
57.5 
58.5 
59.4 
60.2 
61.1 
61.9 
62.6 
63.4 
64.2 
65.1 
66.0 
66.8 
67.6 
68.5 
69.2 

12.5 
13.6 
14.6 
15.4 
16.2 
16.6 
17.1 
17.3 
17.4 
17.5 
17.5 
18.1 
18.8 
19.3 
19.8 
20.2 
20.6 
21.1 
21.4 
21.8 
22.1 
22.7 
23.4 
24.0 
24.6 
25.1 
25.8 
26.2 
26.6 
27.1 
27.5 
27.9 
28.3 
28.6 
29.0 
29.3 
29.6 
29.9 
30.2 
30.6 
31.1 
31.5 
31.9 
32.3 

50.0 

54.5 
58.4 
61.6 
65.2 
68.3 
71.1 
73.5 
75.9 
78.6 
81.9 
84.9 
87.6 
90.2 
92.4 
94.6 
97.1 
100.1 
102.8 
105.4 
107.9 
110.6 
113.7 
116.7 
119.4 
122.0 
124.4 
126.9 
129.7 
132.3 
135.0 
137.6 
140.2 
142.9 
145.6 
148.3 
150.9 
153.4 
156.0 
158.5 
161.0 
163.6 
166.6 
169.6 

11  
12  

13  
14  .  ... 

15  

16  
17  
18  
19  
20  
21  
22  

23  
24  

25  

28  
27  
28  
29  
30  
31  
32  

33  
34  
35  
36  
37  
38  
39  
40  

41 

42  . 

43  
44  
45  
46  
47  
48  
49  
50 

51.  ... 
52  
53  

92  LIVE-LOAD    STRESSES 

TABLE  7. — Continued 

MAXIMUM  MOMENTS,   SHEARS,    AND  PIER  REACTIONS  FOR  COOPER'S 
STANDARD  LOADINGS 

(Figures  for  One  Rail) 


Span 

#40 

£50 

Max. 
Moment 

Max.  Shears 

Max. 
Pier 
React. 

Max. 
Moment 

Max.  Shears 

Max. 
Pier 
React. 

End 

&Pt. 

Cent. 

End 

KPt. 

Cent. 

54.. 
55.  . 

1081.4 
1116.9 
1152.4 
1187.9 
1223.4 
1261.0 
1299.6 
1338.3 
1377.0 
1415.6 
1455.5 
1497.5 
1539.5 
1581.5 
1623.5 
1665.5 
1707.5 
1749.3 
1793.0 
1833.9 
1879.2 
1925.8 
1972.0 
2019.1 
2065.0 
2112.3 
2160.5 
2207.7 
2256.7 
2306.5 
2356.3 
2406.9 
2459.6 
2510.6 
2564.2 
2615.9 
2670.5 
2723.0 
2776.7 
2831.5 
2885.3 
2939.5 
2994.5 
3049.0 

91.5 
92.6 
93.7 
94.8 
95.9 
97.0 
98.0 
99.2 
100.1 
101.3 
102.6 
103.8 
105.0 
106.4 
107.8 
109.2 
110.5 
111.8 
113.3 
114.8 
116.3 
117.7 
119.1 
120.4 
121.7 
123.0 
124.2 
125.6 
126.9 
128.2 
129.5 
130.7 
132.1 
133.4 
134.7 
136.0 
137.2 
138.5 
139.8 
141.1 
142.4 
143.6 
144.8 
146.2 

56.1 
56.8 
57.5 

58.2 
58.8 
59.5 
60.1 
60.7 
61.3 
61.8 
62.4 
63.0 
63.6 
64.2 
64.8 
65.4 
65.9 
66.5 
67.0 
67.5 
68.0 
68.6 
69.2 
69.9 
70.5 
71.1 
71.7 
72.3 
73.0 
73.7 
74.4 
75.1 
75.8 
76.5 
77.1 
77.9 
78.7 
79.5 
80.3 
81.0 
81.7 
82.5 
83.3 
84.2 

26.1 
26.4 
26.6 
26.9 
27.2 
27.5 
27.9 
28.2 
28.5 
28.8 
29.1 
29.4 
29.7 
30.0 
30.2 
30.5 
30.7 
31.1 
31.4 
31.7 
32.0 
32.3 
32.6 
32.9 
33.2 
33.4 
33.7 
34.0 
34.4 
34.7 
35.0 
35.3 
35.6 
35.9 
36.2 
36.5 
36.7 
37.0 
37.3 
37.5 
37.8 
38.0 
38.3 
38.5 

138.0 
140.3 
142.7 
145.4 
148.1 
150.6 
153.2 
155.7 
158.2 
160.4 
162.6 
165.2 
167.8 
170.1 
172.5 
174.8 
177.1 
179.3 
181.5 
183.7 
186.  C 
188.2 
190.4 
192.5 
194.7 
196.8 
198  ..9 
200.9 
203.0 
205.0 
206.9 
208.9 
210.8 
212.8 
214.7 
216.7 
218.6 
220.6 
222  5 
224.4 
226.3 
228  .  1 
230.0 
231.8 

1351.8 
1396  .  1 
1440.5 
1484.9 
1529.2 
1576.2 
1624.5 
1672.9 
1721.2 
'1769.5 
1819.4 
1871.9 
1924.4 
1976.9 
2029.4 
2081.9 
2134.4 
2186.6 
2241.2 
2292.4 
2349.0 
2407.3 
2465.0 
2523.9 
2581.2 
2640.4 
2700.6 
2759.6 
2820.9 
2883  .  1 
2945.4 
3008.6 
3074.5 
3138.3 
3205.3 
3269.9 
3338  .  1 
3403  .  7 
3470.9 
3539  .  3 
3606.6 
3674  .  3 
3743  .  1 
3811.2 

114.5 
115.8 
117.2 
118.5 
119.8 
121.2 
122.5 
123.9 
125.2 
126.6 
128.2 
129.7 
131.2 
133.0 
134.8 
136.5 
138.1 
139.8 
141.7 
143.5 
145.3 
147.1 
148.8 
150.5 
152.1 
153.8 
155.3 
157.0 
158.6 
160.3 
161.8 
163.4 
165.1 
166.8 
168.4 
170.0 
171.5 
173.1 
174.7 
176.4 
178.0 
179.5 
181.0 
182.7 

70.1 
71.0 

71.8 
72.7 
73.5 
74.4 
75.2 
76.0 
76.6 
77.4 
78.0 
78,8 
79.5 
80.3 
81.0 
81.7 
82.4 
83.1 
83.8 
84.4 
85.0 
85.7 
86.5 
87.4 
88.2 
88.9 
89.6 
90.4 
91.2 
92.1 
93.0 
93.9 
94.3 
95.7 
96.5 
97.4 
98.4 
99.4 
100.4 
101.2 
102.1 
103.1 
104.1 
105.1 

32.6 
33.0 
33.3 
33.6 
34.0 
34.4 
34.9 
35.2 
35.6 
36.0 
36.4 
36.8 
37.1 
37.5 
37.8 
38.1 
38.4 
38.8 
39.2 
39.6 
40.0 
40.4 
40.8 
41.1 
41.5 
41.7 
42.1 
42.5 
43.0 
43.4 
43.7 
44.1 
44.5 
44.9 
45.2 
45.6 
45.9 
46.2 
46.6 
46.9 
47.3 
47.5 
47.9 
48.1 

172.5 

175.4 
178.5 
181.8 
185.1 
188.4 
191.5 
194.7 
197.7 
200.7 
203.6 
206.7 
209.7 
212.7 
215.6 
218.5 
221.3 
224.1 
226.9 
229.6 
232.4 
235.2 
238.0 
240.7 
243.3 
245.9 
248.6 
251.1 
253.6 
256.1 
258.7 
260.8 
263.0 
265.6 
268.3 
270.8 
273.2 
275.6 
278.0 
280.3 
282.7 
285.1 
287.5 
2F9.7 

56  

57  

58 

59 

60 

61.  . 

62.  .    . 

63  
64  
65  
66 

67 

68  
69  

70  

71  

72 

73 

74.  . 

75.  . 

76  

77  
78  

79  

80.  . 

81  

82  

83  

84 

85. 

86.  . 

87.. 

88  

89  

90  
91 

92  
93  
94.  .. 

95  
96  
97 

LIVE-LOAD    STRESSES  93 

TABLE  7.— Continued 

MAXIMUM  MOMENTS,   SHEARS  AND  PIER  REACTIONS  FOR  COOPER'S 
STANDARD  LOADINGS 

(Figures  for  One  Rail) 


Span 

E40 

£750 

Max. 
Moment 

Max.  Shears 

Max. 
Pier 
React. 

Max. 
Moment 

Max.  Shears 

Max. 
Pier 
React. 

End 

MPt. 

Cent. 

End 

HPt. 

Cent. 

98    .     ... 

3106.5 
3162.3 
3219.9 
3277.6 
3335.9 
3410.6 
3475.2 
3537.6 
3600.3 
3666.6 
3745.3 
3818.4 
3886.8 
3958.2 
4026.9 
4099.0 
4172.0 
4245.0 
4318.8 
4389.5 
4463.8 
4538.8 
4614.1 
4686.5 
4762.7 
4836.2 
4917.4 
4996.4 
7062.3 
9352.5 
11873.0 
17592.5 

147.5 
148.8 
150.0 
151.2 
152.4 
153.7 
154.9 
156.1 
157.3 
158.5 
159.6 
160.8 
162.0 
163.2 
164.4 
165.5 
165.7 
167.9 
169.0 
170.2 
171.4 
172.5 
173.7 
174.8 
176.0 
177.1 
178.3 
179.4 
207.4 
234.5 
261.0 
313.2 

85.0 
85.8 
86.6 
87.3 
88.1 
88.8 
89.5 
90.3 
90.9 
91.7 
92.4 
93.2 
93.9 
94.6 
95.3 
96.0 
98.8 
97.5 
98.3 
99.0 
99.7 
100.4 
101.1 
101.8 
102.5 
103.2 
104.0 
104.7 
121.8 
138.3 
153.4 
183.7 

38.8 
39.1 
39.4 
39.6 
39.9 
40.1 
40.4 
40.6 
40.9 
41.1 
41.3 
41.6 
41.8 
42.0 
42.2 
42.5 
42.8 
43.1 
43.4 
43.7 
43.9 
44.2 
44.5 
44.7 
45.0 
45.3 
45.7 
46.0 
54.4 
62.5 
70.4 
85.0 

233.6 
235.4 
237.2 
238.9 
240.6 
242.4 
244.2 
246.0 
247.8 
249.6 
251.4 
253.1 
254.8 
256.5 
258.2 
259.9 
281.6 
263.3 
264.9 
266.7 
268.5 
270.2 
272.0 
273.8 
275.6 
277.4 
279.2 
281.0 
325.4 
371.7 
419.0 
515.2 

3883.1 
3952.9 
4024.9 
4097.0 
4169.9 
4263.3 
4344.0 
4422.0 
4500.4 
4583.3 
4681.6 
4773.0 
4858.5 
4947.7 
5033.6 
5123.8 
5215.0 
5306.2 
5398.5 
5486.9 
5579.7 
5673.5 
5767.6 
5858.1 
5953.4 
6045.2 
6146.7 
6245.5 
8827.9 
11690.6 
14841.2 
21990.6 

184.3 
186.0 
187.5 
189.0 
190.6 
192.1 
193.6 
195.1 
196.6 
198.1 
199.5 
201.0 
202.5 
204.0 
205.5 
207.0 
208.4 
209.9 
211.3 
212.8 
214.2 
215.7 
217.1 
218.6 
220.0 
221.4 
222.8 
224.2 
259.2 
293.1 
326.3 
391.5 

106.2 
107.2 
108.2 
109.1 
110.1 
111.0 
111.9 
112.7 
113.6 
114.5 
115.5 
116.4 
117.4 
118.2 
119.1 
120.0 
121.0 
121.9 
122.9 
123.7 
124.6 
125.5 
126.4 
127.2 
128.1 
129.0 
130.0 
130.9 
152.2 
172.9 
191.8 
229.6 

48.5 
48.9 
49.2 
49.5 
49.9 
50.1 
50.5 
50.7 
51.1 
51.5 
51.7 
52.0 
52.3 
52.5 
52.7 
53.1 
53.5 
53.9 
54.2 
54.6 
54.9 
55.3 
55.6 
55.9 
56.2 
56.5 
57.0 
57.5 
68.0 
78.2 
88.0 
106.3 

292.0 
294.2 
296.5 
298.6 
300.8 
303.0 
305.3 
307.5 
309.8 
312.0 
314.2 
316.3 
318.5 
320.7 
322.8 
324.9 
327.0 
329.0 
331.1 
333.3 
335.6 
337.8 
340.0 
342.2 
344.5 
346.7 
349.0 
351.2 
406.7 
464.6 
523.8 
644.0 

99  
100  
101  
102  

103  
104  
105  

106  

107  

108 

109  

110 

111  

112  . 

113 

114  
115  
116  
117  
118  
119  
120  

121  

122  

123  

124  
125  
150  
175  
200 

250  

NOTES. — Moments  are  given  in  thousand  foot-pounds. 
Shears  are  given  in  thousand  pounds. 

Pier  reactions  are  given  in  thousand  pounds  and  are  for  piers  between  two  spans  each  equal 
to  the  tabulated  span. 


94 


LIVE-LOAD    STRESSES 


TABLE  8 

MAXIMUM  MOMENTS  FOR  TRUSS  BRIDGES — COOPER'S  #50  FOR  ONE  RAIL 
Moments  Given  in  Thousands  of  Foot-Pounds 


Panel  Points  h 


Panels 
in  Truss 

1? 

ll 

PANEL  LENGTHS 

8'0" 

8'  6" 

9'0" 

9'  6" 

10'  0" 

10'  6" 

11'  0" 

11'  6" 

12'  0" 

12'  6" 

13'  0" 

13'  6" 

3 

1 

325 

359 

392 

425 

464 

503 

541 

580 

619 

661 

707 

755 

4 

1 

2 

433 
569 

483 
625 

533 
683 

582 
747 

632 
819 

688 
892 

743 
964 

799 

1037 

859 
1110 

918 
1189 

982 
1269 

1046 
1352 

5 

1 
2 

540 
790 

599 

877 

662 
964 

728 
1051 

794 
1149 

861 
1255 

930 
1361 

1001 
1468 

1071 
1574 

1140 
1675 

1217 
1792 

1298 
1910 

6 

1 
2 
3 

641 
1008 
1109 

710 
1115 
1221 

784 
1228 
1351 

859 
1347 
1484 

937 
1466 
1618 

1017 
1587 
1767 

1100 
1719 
1925 

1186 
1857 
2070 

1280 
1997 
2240 

1375 
2135 
2407 

1485 
2289 
2581 

1600 
2451 
2760 

7 

1 
2 
3 

731 
1215 
1425 

812 
1344 
1577 

896 
1477 
1739 

984 
1615 
1910 

1080 
1758 
2086 

1184 
1904 
2269 

1293 
2070 
2465 

1411 
2252 
2667 

1530 
2441 
2879 

1645 
2642 
3100 

1775 
2849 
3332 

1906 
3050 
3560 

8 

1 
2 
3 
4 

819 
1402 
1716 
1819 

915 
1553 
1899 
2030 

1021 
1709 
2100 
2240 

1133 
1872 
2311 
2465 

1254 
2061 
2529 
2700 

1375 
2273 
2752 
2946 

1501 
2490 
2991 
3205 

1631 
2708 
3241 
3471 

1776 
2933 
3498 
3743 

1900 
3165 
3775 
4025 

2047 
3405 
4078 
4344 

2200 
3649 
4383 
4681 

9 

1 
2 
3 
4 

621 
1583 
1997 
2208 

1039 
1764 
2215 
2459 

1162 
1960 
2451 
2719 

1287 
2179 
2700 
2997 

1418 
2405 
2986 
3291 

1556 
2642 
3276 
3592 

1697 
2888 
3570 
3899 

1844 
3139 
3877 
4226 

1997 
3400 
4194 
4588 

2145 
3670 
4532 
4970 

2309 
3946 
4887 
5370 

2475 
4224 
5242 
5770 

l! 

!•! 

££ 

PANEL  LENGTHS 

14'  0" 

14'  6" 

15'  0" 

15'  6" 

16'  0" 

16'  6" 

17'  0" 

17'  6" 

18'  0" 

18'  6" 

19'  0" 

3 

i 

803 

850 

900 

952 

1008 

1060 

1115 

1170 

1228 

1285 

1347 

4 

i 

2 

1115 
1441 

1183 
1529 

1255 
1624 

1325 
1721 

1402 
1820 

1463 
1924 

1553 
2030 

1614 
2134 

1709 
2240 

1776 
2349 

1872 
2465 

5 

.  1 
2 

1389 
2047 

1480 
2177 

1581 
2310 

1680 
2440 

1788 
2581 

1896 
2725 

2010 
2881 

2123 
3030 

2242 
3190 

2355 
3350 

2477 
3518 

6 

1 
2 
3 

1724 
2616 
2946 

1840 
2792 
3138 

1965 
2986 
3338 

2090 
3175 
3539 

2221 
3372 
3742 

2352 
3570 
3953 

2489 
3775 
4170 

2626 
3978 
4422 

2769 
4194 
4681 

2910 
4415 
4948 

3062 
4650 
5215 

7 

1 
2 
3 

2047 
3263 
3802 

2185 
3485 
4040 

2332 
3723 
4310 

2480 
3958 
4595 

2634 
4202 
4898 

2787 
4450 
5200 

2945 
4705 
5509 

3104 
4958 
5815 

3268 
5218 
6135 

3434 
5480 
6460 

3605 
5748 
6800 

8 

1 
2 

3 
4 

2358 
3900 
4710 
5034 

2516 
4165 
5040 
5398 

2681 
4436 
5380 
5768 

2846 
4710 
5720 
6147 

3019 
4994 
6072 
6516 

3190 
5280 
6430 
6915 

3372 
5576 
6806 
7331 

3553 
5873 
7180 
7740 

3741 
6180 
7573 
8163 

3930 

6487 
7985 
8595 

4125 
6805 
8369 
9043 

9 

1 
2 
3 
4 

2651  2828 
4512  4804 
5617*  5993 
6187  6610 

3012 
5107 
6390 
7040 

3196 
5420 
6790 
7485 

3389 

5747 
7204 
7966 

3583 
6074 
7620 
6460 

3785 
6414 
8054 
8980 

3987 
6755 
8496 
9490 

4198 
7108 
8959 
10010 

4410 
7463 
9415 
10530 

4629 
7830 
9892 
11065 

LIVE-LOAD    STRESSES 


95 


TABLE   8.— Continued 

MAXIMUM  MOMENTS  FOR  TRUSS  BRIDGES — COOPER'S  E50  FOR  ONE  RAIL 
Moments  Given  in  Thousands  of  Foot-Pounds 


Panel  Points  h 


4- 


1 


ll 

(5.2 
3 

1 

1 

PANEL  LENGTHS 

19'  6" 

20'  0" 

20'  6" 

21'  0" 

21'  6" 

22'  0" 

22'  6" 

23  '  0" 

23'  6" 

24'  0" 

24'  6" 

1404 

1466 

1527 

1587 

1653 

1719 

1788 

1857 

1927 

1997 

2066 

4 

1 
2 

1958 
2581 

2061 
2700 

2166 
2821 

2273 
2946 

2380 
3074 

2490 
3205 

2597 
3338 

2708 
3471 

2819 
3607 

2933 
3743 

3046 
3883 

5 

1 
2 

2600 
3685 

2731 
3943 

2864 
4144 

3001 
4347 

3138 
4555 

3279 

4767 

3418 
4978 

3562 
5193 

3705 
5415 

3852 
5640 

3999 
5865 

6 

1 

2 
3 

3210 

4885 
5487 

3362 
5256 
5746 

3516 
5501 
6028 

3678 
5750 
6321 

3840 
5998 
6617 

4008 
6250 
6921 

4175 
6501 
7228 

4349 
6756 
7538 

4522 
7011 
7850 

4700 
7270 
8166 

4878 
7525 
8491 

7 

1 
2 
3 

3778 
6025 
7140 

3955 
6326 
7646 

4130 
6613 
7990 

4317 
6914 
8347 

4505 
7215 
8710 

4702 
7530 
9079 

4897 
7845 
9448 

5100 
8173 
9826 

5303 
8503 
10207 

5512 
8842 
10609 

5721 
9182 
11017 

8 

1 
2 
3 

4 

4320 
7125 
8780 
94,0 

4525 
7458 
9234 
9943 

4727 
7805 
9330 
10396 

4939 
8162 
10070 
10862 

5150 
8520 
10515 
11317 

5373 
8890 
10993 
11805 

5592 
9260 
11475 
12288 

5829 
9640 
11976 
12790 

6061 
10030 
12472 
13287 

6300 
10430 
12981 
13795 

6540 
10832 
13490 
14300 

9 

1 
2 
3 

4 

4850 
8198 
10372 
11605 

5379 

8578 
10880 
12172 

5308 
8970 
11375 
12735 

5545 
9378 
11900 
13310 

5780 
9790 
12425 
13880 

6030 
10216 
12978 
14472 

6280 
10640 
13535 
15068 

6542 
11082 
14118 
15684 

6804 
11525 
14705 
16300 

7074 
11985 
15308 
16930 

7344 
12448 
15910 
17560 

I 

•12 
I.S 

ll 

12 

PANEL  LENGTHS 

25'  0" 

25'  6" 

26'  0" 

26'  6" 

27'  0" 

27'  6" 

28'  0" 

28'  6" 

29'  0" 

29'  6" 

30'  0" 

3 

1 

2135 

2215 

2289 

2370 

2451 

2534 

2616 

2700 

2792 

2889 

2986 

4 

1 
2 

3165 
4025 

3282 
4170 

3405 

4344 

3526 
4501 

3649 
4681 

3774 
4858 

3900 
5034 

4031 
5215 

4165 
5398 

4300 
5580 

4436 
5768 

5 

1 
2 

4150 
6093 

4301 
6371 

4456 
6552 

4611 
6783 

4770 
7017 

4929 
7250 

5092 
7492 

5255 
7736 

5422 
7984 

5589 
8232 

5760 
8482 

6 

1 
2 
3 

5061 
7794 
8821 

5245 
8088 
9153 

5433 
8352 
9490 

5622 
8654 
9828 

5816 
8960 
10170 

6010 
9268 
10514 

6208 
9580 
10862 

6408 
9897 
11208 

6612 
10218 
11565 

6817 
10547 
11925 

7026 
10880 
12296 

7 

1 
2 
3 

5936 
9530 
11444 

6151 
9875 
11870 

6373 
10236 
12312 

6595 
10600 
12752 

6823 
10980 
13203 

7051 
11357 
13653 

7286 
11742 
14112 

7521 
12125 
14571 

7762 
12520 
15039 

8003 
12918 
15507 

8250 
13330 
15984 

8 

1 

2 
3 
4 

6787 
11244 
14010 
14820 

7035 
11655 
14528 
15340 

7289 
12080 
15063 
15875 

7540 
12508 
15605 
16413 

7806 
12950 
16163 
16965 

8069 
13392 
16718 
17514 

8338 
13850 
17285 
18075 

8608 
14308 
17852 
18635 

8887 
14780 
18431 
19210 

9165 
15250 
19010 
19795 

9450 
15730 
19600 
20406 

9 

1 
2 
3 
4 

7622 
12925 
16528 
18205 

7900 
13400 
17145 
18850 

8188 
13890 
17778 
19515 

8477 
14380 
18414 
20180 

8774 
14888 
19070 
20870 

9070 
15400 
19730 
21557 

9376 
15930 
20405 
22260 

9686 
16460 
21080 
22955 

9996 
17005 
21770 
23678 

10310 
17547 
22461 
24405 

10633 
18100 
23168 
25170 

96 


LIVE-LOAD    STRESSES 


TABLE   8.— Continued 

MAXIMUM  MOMENTS  FOR  TRUSS  BRIDGES — COOPER'S  £"50  FOR  ONE  RAIL 
Moments  Given  in  Thousands  of  Foot-Pounds 


Panel  Points  I 


Panels  1 
in  Truss 

ll 

&& 

PANEL  LENGTHS 

30'  6" 

31'  0" 

31'  6" 

32'  0" 

32'  6" 

33'  0" 

33'  6" 

34'  0" 

34'  6" 

35'  0" 

35'  6" 

3 

1 

3080 

3175 

3276 

3372 

3471 

3570 

3672 

3775 

3877 

3978 

4080 

4 

1 

2 

4573 
5957 

4710 
6147 

4852 
6332 

4994 
6516 

5137 
6715 

5280 
6915 

5428 
7123 

5576 
7331 

5725 
7535 

5873 
7740 

5923 
7950 

5 

1 
2 

5937 
8734 

6113 
8986 

6295 
9241 

6477 
9496 

6678 
9749 

6849 
10012 

7039 
10291 

7228 
10590 

7423 
10891 

7617 
11192 

7814 
11495 

6 

1 
2 
3 

7238 
11219 
12668 

7450 
11558 
13040 

7671 
11903 
13418 

7892 
12248 
13796 

8120 
12684 
14180 

8347 
12979 
14563 

8581 
13354 
14952 

8812 
13729 
15341 

9050 
14120 
15745 

9288 
14510 
16148 

9628 
14902 
16654 

7 

1 
2 
3 

8501 
13748 
16474 

8752 
14165 
16964 

9009 
14590 
17466 

9266 
15015 
17968 

9536 
15460 
18475 

9806 
15885 
18981 

10081 
16358 
19508 

10355 
16810 
20015 

10637 
17284 
20545 

10919 
17758 
21024 

11203 
18234 
21606 

8 

1 
2 
3 
4 

9740 
16225 
20206 
21022 

10030 
16720 
20812 
21638 

10326 
17227 
21432 
22268 

10622 
1^33 
22051 
22898 

10931 
18252 
22685 
23549 

11239 
18770 
23318 
24200 

11557 
19311 
23960 
24860 

11874 
19852 
24601 
25531 

12200 
20407 
25261 
26216 

12526 
20961 
25920 
26901 

12856 
21518 
26585 
27590 

9 

1 
2 
3 

4 

10961 
18672 
23886 
25943 

11288 
19244 
24603 
26715 

11625 
19832 
25343 

27498 

11961 
20419 
26083 
28281 

12310 
21019 
26839 
29096 

1265C 
21618 
27595 
29910 

13018 
22239 
28365 
30741 

13378 
22860 
29135 
31572 

13747 
23503 
29923 
32431 

14116 
24146 
30710 
33290 

14490 
24795 
31500 
34155 

LIVE-LOAD    STRESSES 


97 


TABLE  9 

MAXIMUM  SHEARS  FOE  TRUSS  BRIDGES — COOPER'S  #50  FOE  ONE  RAIL 
Shears  Given  in  Thousands  of  Pounds 


Panels 


•d 

"3 

PANEL  LENGTHS 

l« 

c 

2 

8'  0" 

8'  6" 

9'  0" 

9'  6" 

10'  0" 

10'  6" 

11'  0" 

11'  6" 

12'  0" 

12'  6" 

13'  0" 

13'  6" 

3 

i 

40.6 

42.1 

43.5 

44.8 

46.4 

47.9 

49.1 

50.4 

51.6 

53.0 

54.3 

55.9 

2 

7.3 

8.0 

8.8 

9.5 

10.0 

11.0 

11.8 

12.5 

13.2 

13.7 

14.3 

14.9 

4 

1 

54.1 

56.7 

59il 

61.3 

63.1 

65.5 

67.4 

69.4 

71.6 

73.6 

75.5 

77.6 

2 

23.5 

25.4 

27.4 

28.6 

30.0 

31.3 

32.4 

33.4 

34.4 

35.6 

36.7 

37.7 

8 

2.4 

3.1 

3.9 

4.5 

5.0 

5.9 

6.5 

7.2 

7.9 

8.4 

8.9 

9.4 

5 

1 

675 

70.4 

73.6 

76.6 

79.4 

82.3 

84.5 

87.1 

89.2 

91.4 

93.6 

96.4 

2 

38.8 

41.0 

43.0 

44.9 

46.7 

48.7 

50.3 

51.9 

53.8 

55.5 

57.1 

58.7 

3 

16.3 

18.0 

19.5 

20.8 

22.0 

23.1 

24.0 

25.0 

25.9 

26.9 

27.8 

28.7 

6 

1 

80.1 

83.5 

86.9 

90.1 

93.6 

96.9 

100.1 

103.1 

106.7 

110.5 

114.3 

118.7 

2 

52.7 

55.3 

57.9 

60.5 

62.9 

65.5 

67.8 

70.1 

72.1 

74.2 

76.3 

78.1 

8 

30.2 

33.5 

34.0 

35.6 

37.4 

39.0 

40.8 

41.9 

43.4 

44.9 

46.3 

47.7 

4 

11.5 

13.0 

14.4 

15.6 

16.6 

17.8 

18.8 

19.4 

20.2 

21.1 

21.9 

22.6 

7 

1 

91.1 

94.6 

99.2 

103.4 

108.0 

112.8 

1175 

122.9 

127.5 

132.0 

136.5 

141.4 

2 

65.5 

69.1 

72.4 

753 

78.4 

80.9 

83.9 

86.1 

89.0 

920 

95.0 

98.8 

8 

43.4 

45.6 

48.0 

50.4 

52.4 

54.8 

56.9 

58.8 

596 

62.0 

64.3 

65.9 

4 

24.1 

26.0 

27.6 

29.0 

30.5 

32.1 

33.4 

34.7 

36.1 

37.4 

38.6 

39.8 

5 

8.5 

9.6 

10.7 

11.7 

12.8 

13.8 

14.9 

15.5 

16.1 

16.9 

17.7 

18.4 

8 

1 

101.9 

107.6 

113.6 

119.3 

125.4 

131.0 

136.4 

141.9 

147.2 

152.3 

157.4 

162.9 

2 

78.2 

81.7 

85.2 

89.1 

92.5 

96.0 

99.8 

104.1 

108.4 

112.6 

116.7 

121.0 

3 

55.8 

59.0 

61.9 

64.5 

67.4 

69.6 

72.3 

74.4 

76.8 

79.5 

82.2 

85.0 

4 

36.4 

38.5 

40.6 

42.8 

44.6 

46.8 

48.6 

50.4 

52.0 

53.7 

55.3 

56.7 

6 

19.5 

21.3 

22.8 

24.1 

25.5 

26.9 

28.0 

29.1 

30.5 

31.7 

32.8 

33.9 

6 

7.4 

7.9 

8.4 

9.2 

10.0 

10.9 

11.9 

12.5 

13.1 

13.8 

14.5 

15.1 

9 

1 

115.2 

122.3 

129.2 

135.6 

141.9 

148.4 

154.5 

160.8 

1664 

172.0 

177.6 

183.5 

2 

89.0 

93.6 

98.3 

103.3 

108.3 

113.6 

118.6 

123.4 

128.2 

132.9 

137.5 

142.5 

3 

68.1 

71.4 

74.5 

77.6 

81.2 

84.3 

87.8 

91.6 

95.4 

99.2 

102.9 

106.4 

4 

48.2 

51.1 

53.8 

56.5 

58.5 

60.8 

63.1 

65.1 

67.4 

69.8 

72.2 

74.8 

5 

31.0 

32.9 

34.9 

36.9 

38.5 

40.5 

42.3 

43.8 

45-3 

46.8 

48.3 

49.6 

6 

16.0 

17.5 

19.1 

20.3 

21.5 

22.7 

23.9 

25.0 

26.2 

27.3 

28.3 

29.3 

a 

PANEL,  LENGTHS 

fir* 

1 

£.3 

2 

14'  0" 

14'  6" 

15'  0" 

15'  6" 

16'  0" 

16'  6" 

17'  0" 

17'  6" 

18'  0" 

18'  6" 

19'  0" 

3 

i 

57.4 

58.7 

60.0 

615 

63.0 

64.3 

65.6 

66.9 

68.2 

69.5 

70.8 

2 

15.5 

16.0 

16.4 

17.1 

17.8 

18.3 

188 

19.3 

19.9 

20.5 

21.0 

4 

1 

79.6 

81.6 

83.6 

85.5 

87.3 

89.0 

90.6 

92.6 

94.5 

96.4 

98.3 

2 

38.6 

39.6 

40.6 

41.7 

42.7 

43.9 

45.0 

46.1 

47.2 

48.3 

49.3 

8 

9.8 

10.3 

10.7 

11.2 

11.7 

12.2 

12.7 

13.1 

13.5 

13.9 

14.3 

5 

1 

99.2 

102.3 

105.4 

108.6 

111.8 

115.1 

118.3 

121.5 

124.6 

127.5 

130.4 

2 

60.3 

61.9 

63.4 

64.8 

662 

67.7 

69.1 

70.8 

72.4 

74.0 

75.6 

3 

29.5 

30.4 

31.2 

320 

328 

33.6 

34.3 

351 

35.8 

36.6 

37.3 

6 

1 

123.1 

127.1 

131.0 

134.9 

138.8 

142.7 

146.5 

1502 

153.8 

157.5 

161.1 

2 

79.8 

82.2 

84.6 

86.9 

90.1 

93.0 

95.8 

98.5 

101.1 

103.6 

106.1 

8 

49.1 

50.4 

51.7 

52.9 

54.0 

55.3 

56.5 

57.6 

58.6 

59.7 

60.7 

4 

23.3 

24.1 

24.8 

25.6 

26.3 

27.0 

27.6 

28.3 

28.9 

29.6 

30.2 

7 

1 

146.2 

150.9 

155.5 

160.1 

164.6 

169.0 

173.3 

177.5 

181.6 

185.7 

189.7 

2 

102.6 

106.1 

109.6 

113.0 

116.4 

119.7 

123.1 

126.4 

129.6 

132.8 

135.9 

8 

67.4 

69.3 

71.1 

73.1 

75.0 

77.4 

79.7 

82.1 

84.4 

86.6 

88.8 

4 

41.0 

42.2 

43.4 

44.4 

45.4 

46.5 

47.5 

48.5 

49.4 

50.4 

51.3 

5 

19.0 

19.7 

203 

21.0 

21.6 

22.2 

22.8 

23.4 

24.0 

24.6 

25.1 

8 

1 

168.4 

173.6 

178.8 

183.8 

188.7 

193.6 

198.4 

203.1 

207.8 

212.5 

217.1 

2 

125.3 

129.5 

133.7 

1378 

141.8 

145.7 

149.5 

153.2 

156.9 

160.5 

164.1 

8 

87.8 

90.9 

93.9 

96.8 

99.6 

102.6 

105.6 

108.5 

111.4 

114.2 

117.0 

4 

58.1 

59.8 

61.4 

63.1 

64.8 

66.7 

68.5 

70.4 

72.2 

74.0 

75.8 

6 

35.0 

36.1 

37.1 

38.0 

38.9 

39.9 

40.9 

41.7 

42.5 

43.4 

44.2 

6 

15.7 

16.4 

17.0 

17.6 

18.1 

18.7 

19.2 

19.8 

20.3 

20.8 

21.3 

9 

189.4 

195.1 

200.8 

206.3 

211.8 

217.3 

222.7 

2280 

233.2 

238.4 

243.6 

2 

147.4 

152.1 

156.8 

1613 

165.7 

170.1 

174.5 

178.8 

183.0 

187.2 

191.3 

3 

109.8 

112.9 

116.7 

120.4 

124.1 

127.6 

131.0 

134.4 

137.7 

141.0 

144.2 

4 

77.3 

80.1 

82.7 

85.2 

87.6 

90.1 

92.5 

94.9 

97.3 

99.9 

102.4 

5 

50.8 

52.4 

53.8 

55.4 

56.9 

58.6 

60.2 

61.9 

63.5 

65.3 

67.0 

6 

30.3 

31.4 

32.3 

33.1 

33.9 

34.8 

35.7 

36.5 

37.2 

38.0 

38.7 

98 


LIVE-LOAD    STRESSES 


TABLE   9. — Continued 

MAXIMUM  SHEARS  FOR  TRUSS  BRIDGES — COOPER'S  E5Q  FOR  ONE  RAIL 
Shears  Given  in  Thousands  of  Pounds 


1              2               3               4.5.6,7.8.9. 
Panels           I  =  1  1  1  1  «  

PANEL  LENGTHS 

|| 

1 

19'  6" 

20'  0" 

20'  6" 

21'  0" 

21'  6" 

22'  0" 

22'  6" 

23'  0" 

23'  6" 

24'  0" 

24'  6" 

3 

1 

72.0 

73.3 

74.3 

75.3 

76.6 

78.0 

79.5 

81.0 

82.1 

83.2 

84.6 

2 

21.5 

22.0 

22.4 

22.9 

23.5 

24.0 

24.3 

24.6 

25.1 

25.5 

25.9 

4 

1 

100.7 

103.0 

105.6 

108.2 

110.7 

113.2 

115.5 

117.7 

120.0 

122.2 

124.4 

2 

50.3 

51.3 

52.2 

53.1 

54.0 

54.9 

55.8 

56.8 

57.4 

58.2 

59.0 

3 

14.7 

15.0 

15.3 

15.6 

15.9 

16.2 

16.5 

16.7 

17.0 

17.2 

17.5 

5 

1 

133.5 

136.6 

139.8 

142.9 

146.0 

149.0 

152.0 

154.9 

157.8 

160.5 

163.3 

2 

77.4 

79.1 

80.9 

82.6 

84.4 

86.1 

88.0 

89.9 

91.7 

93.5 

95.1 

3 

38.1 

38.8 

39.6 

40.3- 

40.9 

41.6 

42.3 

42.9 

43.7 

44.3 

45.0 

6 

1 

164.6 

168.1 

171.7 

175.2 

178.8 

182.3 

185.8 

189.2 

192.6 

195.9 

199.2 

2 

108.6 

111.0 

113.6 

116.0 

118.5 

120.8 

123.2 

125.4 

127.9 

130.1 

132.4 

3 

62.1 

63.5 

65.1 

66.6 

68.2 

69.6 

71.3 

72.9 

74.5 

75.9 

77.4 

4 

30.8 

31.4 

32.1 

32.8 

33.4 

34.0 

34.5 

35.0 

35.5 

36.0 

36.6 

7 

1 

193.9 

197.8 

201.7 

205.5 

209.6 

213.7 

217.8 

221.8 

225.8 

229.7 

233.6 

2 

139.0 

142.0 

145.0 

147.9 

150.9 

153.7 

156.1 

159.3 

162.1 

164.8 

167.6 

3 

91  0 

93  1 

95  4 

97  5 

99  6 

101  6 

103  8 

105.8 

107.9 

109.8 

111    8 

4 

52.4 

53.4 

54.5 

55.5 

56.7 

57.8 

59.3 

60.6 

62.1 

63.4 

64.7 

5 

25.7 

26.3 

26.9 

27.4 

28.0 

28.5 

29.0 

29.4 

29.9 

30.3 

30.8 

8 

1 

221.7 

226.3 

230.8 

235.2 

239.8 

244.3 

248.9 

253.4 

258.0 

262.5 

267.1 

2 

167.7 

171.3 

174.8 

178.2 

181.7 

185.0 

188.4 

191.7 

195.1 

198.3 

201.7 

3 

119  8 

122  5 

125  1 

127  6 

130.5 

132.8 

135.4 

137.8 

140.3 

142.7 

145  2 

4 

77.8 

79.8 

81.7 

83.6 

85.5 

87.3 

89.2 

91.0 

92.8 

94.5 

96.3 

5 

45.2 

46.1 

47.1 

48.0 

49.0 

49.4 

51.0 

52.1 

53.1 

54.1 

55.3 

6 

21.9 

22.4 

22.9 

23.4 

23.9 

24.4 

24.9 

25.3 

25.7 

26.0 

26  5 

9 

1 

248.8 

253.9 

259.0 

264.0 

269.2 

274.2 

279.4 

284.5 

289,7 

294.8 

299.9 

2 

195.4 

199.5 

203.5 

207.5 

211.5 

215.5 

219.4 

223.3 

227.2 

231.0 

234.9 

3 

147.4 

150.6 

163.8 

156.9 

160.0 

163.0 

166.0 

169.0 

172.0 

175.0 

177.9 

4 

104.9 

107.3 

109.7 

112.0 

114.3 

116.6 

118.9 

121.1 

123.4 

125.5 

127.8 

5 

68.6 

70.1 

71.7 

73.3 

74.9 

76.4 

78.0 

79.5 

81.2 

82.8 

84.3 

6 

39.6 

40.4 

41.3 

42.1 

43.0 

43.9 

44.9 

45.8 

46.7 

47.6 

48.6 

I 

PANEL  LENGTHS 

rj 

"3 

Is 

25'  0" 

25'  6" 

26'  0" 

26'  6" 

27'  0" 

27'  6" 

28'  0" 

28'  6" 

29'  0" 

29'  6" 

30'  0" 

3 

1 

86.0 

87.0 

88.0 

89.5 

91.0 

92.2 

93.5 

94.7 

96.0 

97.8 

99.7 

2 

26.4 

26.8 

27.2 

27.6 

28.0 

28.3 

28.6 

29.0 

29.4 

29.7 

30.0 

4 

1 

126.5 

128.7 

130.9 

133.1 

135.2 

137.3 

139.3 

141.5 

143.6 

145.8 

147.9 

2 

59.7 

60.5 

61.3 

62.1 

62.9 

63.8 

64.6 

65.6 

66.5 

67.4 

68.3 

3 

17.8 

18.1 

18.4 

18.6 

18.9 

19.1 

19.3 

19.6 

19.8 

20.1 

20.3 

5 

1 

166.0 

168.8 

171.4 

174.1 

176.7 

179.4 

181.9 

184.5 

187.0 

189.6 

192.0 

2 

96.6 

98.3 

100.1 

101.9 

103.6 

105.4 

107.1 

108.9 

110.6 

112.3 

114.0 

3 

45.5 

46.3 

46.9 

47.7 

48.3 

49.0 

49.6 

50.5 

51.3 

52.1 

52.8 

6 

1 

202.5 

205.8 

209.0 

212.2 

215.4 

218.6 

221.8 

224.9 

228.0 

231.1 

234.2 

2 

134.5 

136.8 

139.0 

141.3 

143.5 

145.8 

148.0 

150.3 

152.4 

154.6 

156.7 

3 

78.6 

80.2 

81.5 

83.0 

84.3 

85.7 

87.0 

88.4 

89.6 

91.1 

92.4 

4 

37.1 

37.6 

38.1 

38.6 

39.1 

39.6 

40.0 

40.5 

41.0 

41.7 

42.4 

7 

1 

237.4 

241.4 

245.2 

249.1 

252.8 

256.6 

260.3 

264.1 

267.7 

271.4 

275.0 

2 

170.3 

173.2 

175.9 

178.8 

181.5 

184.3 

187.0 

189.8 

192.5 

195.3 

197.9 

3 

113.6 

115.6 

117.4 

119.3 

121.1 

123.0 

124.8 

126.6 

128.3 

130.2 

131.9 

4 

65.8 

67.1 

68.3 

69.6 

70.8 

72.0 

73.1 

74.3 

75.4 

76.7 

77.8 

5 

31.3 

31.8 

32.1 

32.6 

33.0 

33.5 

33.8 

34.3 

34.6 

35.1 

35.6 

8 

1 

271.5 

276.0 

280.4 

284.9 

289.2 

293.6 

297.9 

302.3 

306.5 

310.8 

315.0 

2 

204.9 

208.3 

211.6 

215.1 

218.4 

221.8 

225.0 

228  A 

231.7 

235.0 

238.2 

3 

147.5 

150.0 

152.3 

154.7 

157.0 

159.4 

161.7 

164.0 

166.1 

168.5 

170.2 

4 

98.0 

99.8 

101.4 

103.1 

104.6 

106.3 

107.9 

109.5 

111.0 

112.6 

114.1 

5 

56.4 

57.4 

58.4 

59.5 

60.5 

61.6 

62.6 

63.7 

'64.8 

65.9 

66.9 

6 

26.9 

27.3 

27.6 

28.0 

28.4 

28.8 

29.1 

29.5 

29.9 

30.4 

30.8 

9 

1 

304.9 

310.0 

315.0 

320.1 

325.0 

330.0 

334.9 

339.9 

344.7 

349.7 

354.5 

2 

238.8 

242.8 

246.7 

250.6 

254.5 

258.5 

262  .4 

266.3 

270.2 

274.0 

277.8 

3 

180.8 

183.8 

186.7 

189.6 

192.4 

195.3 

198.0 

200.9 

203.8 

206.7 

209  .  ? 

4 

129.9 

132.0 

134.1 

136.3 

138  .4 

140.5 

142.5 

144.6 

146.6 

148.6 

150.fi 

5 

85.8 

87.4 

88.9 

90.4 

91.8 

93.3 

94.8 

96.2 

97.6 

99.0 

100.4 

6 

49.6 

50.6 

51.5 

52  A 

53.3 

54.2 

55.0 

55.9 

56.8 

57.6 

58.4 

LIVE-LOAD    STRESSES 


99 


TABLE   9.— Continued 

MAXIMUM  SHEARS  FOR  TRUSS  BRIDGES — COOPER'S  #50  FOR  ONE  RAIL 
Shears  Given  in  Thousands  of  Pounds 


Panels 


Panels 
in  Truss 

1 

PANEL  LENGTHS 

30'  6" 

31'  0" 

31'  6" 

32'  0" 

32'  6" 

33'  0" 

33'  6" 

34'  0" 

34'  6" 

35'  0" 

35'  6" 

3 

1 

101.1 

102.6 

104.6 

106.6 

108.1 

109.6 

111.5 

113.4 

114.8 

116.2 

117.6 

2 

30.4 

30.8 

31.2 

31.5 

31.8 

32.2 

32.5 

32.8 

33.1 

33.4 

33.7 

4 

1 

149.9 

152.0 

154.0 

156.1 

158.0 

160.0 

161.9 

163.8 

165.8 

167.9 

169.8 

2 

69.1 

70.0 

71.7 

73.3 

74.4 

75.4 

76.4 

77.4 

78.4 

79.4 

80.5 

3 

20.6 

20.9 

21.1 

21.3 

21.6 

22.0 

22.2 

22.5 

22.7 

23.0 

23.3 

5 

1 

194.6 

197.1 

199.8 

202.4 

205.0 

207.5 

210.1 

212.6 

215.1 

217.6 

220.2 

2 

115.6 

117.3 

118.9 

120.4 

122.0 

123.5 

125.0 

126.5 

128.0 

129.5 

131.0 

3 

53.6 

54.3 

55.1 

55.9 

56.7 

57.4 

58.3 

59.1 

60.0 

60.8 

61.7 

6 

1 

237.3 

240.3 

243.5 

246.6 

249.8 

252.9 

256.0 

259.1 

262.3 

265.4 

268.5 

2 

158.8 

160.9 

163.0 

165.1 

167.2 

169.3 

171.4 

173.4 

175.4 

177  .4 

179.4 

3 

93.7 

95.0 

96.3 

97.5 

98.8 

100.0 

101.3 

102.5 

103.8 

105.1 

106.4 

4 

43.0 

43.6 

44.4 

45.1 

45.8 

46.4 

47.2 

47.9 

48.6 

49.3 

50.0 

7 

1 

278.7 

282.3 

286.0 

289.6 

293.4 

297.1 

300.9 

304.7 

308.4 

312.0 

315.7 

2 

200.6 

203.3 

205.9 

208.5 

211.2 

213.8 

216.4 

218.9 

221.5 

224.0 

226.5 

3 

133.6 

135.3 

137.1 

138.9 

140.7 

142.5 

144.3 

146.0 

147.9 

149.8 

151.7 

4 

79.0 

80.1 

81.3 

82.4 

83.5 

84.5 

85.6 

86.6 

87.7 

88.7 

89.8 

5 

36.1 

36.5 

37.0 

37.5 

38.0 

38.5 

39.2 

39.9 

40.5 

41.0 

41.6 

8 

1 

319.3 

323.5 

327.8 

332.0 

337.0 

341.9 

345.6 

349.3 

353.2 

357.0 

360.9 

2 

241.4 

244.6 

247.8 

251.0 

254.2 

257.4 

260.6 

263.8 

266.9 

270.0 

273.2 

3 

172.8 

175.4 

177.8 

180.1 

182.5 

184.8 

187.1 

189.4 

191.7 

193.9 

196.2 

4 

115.7 

117.3 

118.7 

120.3 

121.9 

123.4 

124.9 

126.3 

127.7 

129.1 

130.5 

5 

67.9 

68.9 

69.9 

70.9 

71.9 

72.9 

73.9 

74.8 

75.7 

76.6 

77.5 

6 

31.2 

31.5 

32.0 

32.5 

32.9 

33.3 

33.8 

34.3 

34.7 

35.1 

35.5 

9 

1 

359.4 

364.2 

369.1 

373.9 

378.7 

383.5 

388.5 

393.5 

398.4 

403.3 

408.3 

2 

281.6 

285.4 

289.2 

293.0 

296.8 

300.5 

304.3 

308.0 

311.8 

315.5 

319.2 

3 

212.4 

215.3 

218.2 

221.0 

223.9 

226.8 

229.6 

232.5 

235.3 

238.1 

240.8 

4 

152.7 

154.8 

156.8 

158.8 

160.7 

162.6 

164.6 

166.6 

168.6 

170.5 

172  .5 

5 

101.8 

103.1 

104.5 

105.9 

107.3 

108.6 

110.0 

111.4 

112.7 

114.0 

115.4 

6 

59.4 

60.3 

61.2 

62.0 

62.9 

63.8 

64.7 

65.5 

66.3 

67.1 

67.8 

100 


LIVE-LOAD    STRESSES 


TABLE  10 

MAXIMUM  BENDING  MOMENTS  IN  GIRDER  BRIDGES  WITHOUT  FLOOR-BEAMS, 
COOPER'S  E£0  LOADING 


Values  in  Thousands  of  Foot-Pounds  per  Rail 
SHORTER  SEGMENT  h 


5 

10 

15 

20 

25 

30 

35 

40 

45 

50 

55 

60 

250.. 

1534 

3030 

4514 

5979 

7411 

8820 

10203 

11562 

12916 

14278 

15628 

16982 

225.. 

1404 

2769 

4122 

5455 

6758 

8034 

9288 

10515 

11743 

12976 

14198 

15422 

200.  . 

1273 

2505 

3727 

4926 

6098 

7241 

8364 

9460 

10560 

11665 

12759 

13849 

175.. 

1139 

2236 

3326 

4390 

5430 

6438 

7430 

8391 

9364 

10339 

11306112266 

160.. 

1053 

2073 

3082 

4063 

5022 

5950 

6862 

7742 

8638 

9535 

10424 

11300 

150.. 

1003 

1962 

2917 

3843 

4749 

5620 

6480 

7304 

8150 

8994 

9833 

10664 

140.. 

947 

1851 

2750 

3620 

4471 

5287 

6093 

6862 

7658 

8450 

9236 

10016 

130.. 

889 

1738 

2582 

3394 

4191 

4951 

5703 

6417 

7161 

7901 

8635 

9363 

120.. 

834 

1625 

2410 

3164 

3906 

4608 

5307 

5964 

6658 

7345 

8028 

8704 

110.. 

774 

1509 

2234 

2930 

3617 

4260 

4905 

5514 

6148 

6782 

7414 

8038 

M 

100.. 

714 

1390 

2055 

2690 

3320 

3910 

4494 

5053 

5650 

6234 

6813 

7387 

•"-••a 

95.. 

682 

1329 

1963 

2566 

3169 

3730 

4290 

4864 

5431 

5991 

6546 

7096 

1 

90.. 

650 

1264 

1866 

2444 

3016 

3550 

4114 

4661 

5202 

5734 

6263 

6786 

§ 

85.. 

617 

1200 

1770 

2314 

2854 

3365 

3923 

4442 

4936 

5458 

5958 

6449 

bfi 
o 

80.. 

584 

1134 

1671 

2186 

2694 

3200 

3715 

4205 

4690 

5171 

5646 

6117 

CO 

75.. 

551 

1070 

1573 

2054 

2530 

3008 

3489 

3964 

4422 

4874 

5320 

5761 

§3 

70.. 

516 

1003 

1474 

1923 

2366 

2805 

3254 

3706 

4132 

4553 

4967 

5378 

*G 

65.. 

482 

931 

1367 

1792 

2202 

2602 

3019 

3437 

3831 

4221 

4608 

4993 

2 

60.. 

453 

864 

1266 

1649 

2025 

2389 

2770 

3155 

3519 

3884 

4243 

4597 

55.. 

425 

805 

1172 

1518 

1856 

2195 

2546 

2884 

3214 

3514 

3859 

50.  . 

397 

750 

1091 

1398 

1713 

2023 

2336 

2634 

2928 

3219 

45.  . 

367 

692 

1005 

1290 

1567 

1847 

2136 

2404 

2669 

40.  . 

335 

635 

918 

1171 

1419 

1669 

1921 

2160 

35.  . 

302 

570 

819 

1050 

1272 

1490 

1707 

30.. 

270 

506 

721 

918 

1109 

1294 

25.  . 

235 

440 

622 

787 

946 

20.  . 

200 

373 

518 

656 

15.  . 

150 

300 

410 

10.  . 

100 

200 

5.. 

50 

For  h  and  Z2  each  >  142  ft.  M 


LIVE-LOAD    STRESSES 


TABLE  10.— Continued 

MAXIMUM  BENDING  MOMENTS  IN  GIRDER  BRIDGES  WITHOUT  FLOOR-BEAMS, 
COOPER'S  #40  LOADING 

Values  in  Thousands  of  Foot-Pounds  per  Rail 
SHORTER  SEGMENT  l\ 


250 
225 
200 
175 
160 
150 
140 
130 
120 
110 
100 
95 
90 
85 
80 
75 
70 
65 

65 

70 

75 

80 

85 

90 

95 

100 

110 

120 

130 

140 

18327 
16639 
14939 
13224 
12185 
11487 
10790 
10088 
9380 
8666 
7963 
7642 
7303 
6943 
6582 
6197 
5796 
5374 

19675 
17862 
16036 
14205 
13097 
12354 
11608 
10857 
10100 
9338 
8567 
8182 
7817 
7428 
7043 
6629 
6197 

21062 
19123 
17172 
15207 
14018 
13194 
12395 
11594 
10786 
9972 
9150 
8737 
8321 
7917 
7500 
7057 

22421 
20351 
18269 
16171 
14906 
14058 
13206 
12349 
11486 
10616 
9738 
9296 
8851 
8404 
7954 

23766 
21569 
19360 
17134 
15789 
14887 
13980 
13069 
12073 
11226 
10294 
9824 
9352 
8876 

25084 
22757 
20418 
18017 
16636 
15681 
14722 
13756 
12787 
11812 
10829 
10334 
9836 

26364 
23908 
21440 
18952 
17450 
16442 
15430 
14413 
13421 
11392 
11348 
10834 

27660 
25078 
22482 
19868 
18289 
17231 
16169 
15101 
14026 
12946 
11857 

30152 
27315 
24465 
21597 
19866 
18706 
17542 
16372 
15197 
14014 

32591 
29502 
26400 
23278 
21396 
20151 
18870 
17600 
16325 

3503337455 
3169133862 
28231  &0255 
2496326631 
22930)24446 
2156922986 
2020321520 
18834  

1  

For  h  and  12  each  >  142  ft.  M  =  h  I,  +  3800  j~ 

QJ DO  O  OOP  Q  o 


Influence  Line  for  M 


102' 


LIVE-LOAD    STRESSES 


TABLE  11 

MAXIMUM  BENDING  MOMENTS  IN  GIRDER  BRIDGES  WITHOUT  FLOOR-BEAMS, 
COOPER'S,  E50  LOADING 

Values  in  Thousands  of  Foot-Pounds  per  Rail 
SHORTER  SEGMENT  h 


5 

10 

15 

20 

25 

30 

35 

40 

45 

50 

55 

60 

250. 

1918 

3788 

5643 

7474 

9264 

11025 

12754 

14452 

16145 

17848 

19535 

21228 

225. 

1755 

3461 

5153 

6819 

8447 

10043 

11610 

13144 

14679 

16220 

17748 

19278 

200. 

1591 

3131 

4659 

6158 

7622 

9052 

10456 

11825 

13200 

14581 

15949 

17311 

175. 

1424 

2795 

4158 

5487 

6787 

8048 

9288 

10489 

11705 

12924 

14132 

15333 

160. 

1316 

2591 

3852 

5079 

6278 

7437 

8578 

9677 

10798 

11919 

13030 

14125 

150. 

1254 

2453 

3646 

4804 

5936 

7025 

8100 

9130 

10187 

11243 

12291 

13330 

140. 

1184 

2314 

3438 

4525 

5589 

6609 

7617 

8578 

9572 

10562 

11545 

12520 

130. 

1114 

2173 

3227 

4242 

5239 

6189 

7129 

8021 

8951 

9876 

10794 

11704 

120. 

1042 

2031 

3012 

3955 

4883 

5760 

6634 

7455 

8322 

9181 

10035 

10880 

110. 

968 

1886 

2793 

3662 

4521 

5325 

6131 

6892 

7685 

8478 

9268 

10048 

100. 

892 

1737 

2569 

3362 

4150 

4887 

5618 

6316 

7063 

7793 

8516 

9234 

+3 

95. 

853 

1661 

2454 

3208 

3961 

4663 

5363 

6080 

6789 

7489 

8183 

8870 

90. 

812 

1580 

2333 

3055 

3770 

4437 

5143 

5826 

6502 

7168 

7829 

8482 

a 

85. 

771 

1500 

2213 

2893 

3568 

4206 

4904 

5552 

6170 

6823 

7448 

8061 

bC 

rtj 

80. 

730 

1418 

2089 

2733 

3368 

4000 

4644 

5256 

5862 

6464 

7058 

7646 

Qg 

75. 

689 

1337 

1966 

2568 

3163 

3760 

4361 

4955 

5528 

6093 

6650 

7201 

& 

70. 

645 

1254 

1843 

2404 

2958 

3506 

4068 

4632 

5165 

5691 

6209 

6723 

bD 
G 

65. 

602 

1164 

1709 

2240 

2753 

3253 

3774 

4296 

4789 

5276 

5760 

6241 

3 

60. 

566 

1080 

1582 

2061 

2531 

2986 

3463 

3943 

4399 

4855 

5304 

5746 

55. 

531 

1006 

1465 

1897 

2320 

2744 

3182 

3605 

4017 

4392 

4824 



50. 

496 

937 

1364 

1747 

2141 

2529 

2920 

3293 

3660 

4024 

45. 

459 

865 

1256 

1613 

1959 

2309 

2670 

3005 

3336 

40. 

419 

794 

1147 

1464 

1774 

2086 

2401 

2700 

.  .  . 

35. 

377 

713 

1024 

1312 

1590 

1862 

2134 

30. 

338 

632 

901 

1148 

1386 

1617 

25. 

294 

550 

778 

984 

1182 

20. 

250 

466 

647 

820 

15. 

187 

375 

513 

10. 

125 

250 

5. 

62 

For  ^  and  12  each  >  142  ft.  M  =  1.25  I,  L  +  4750  f 


LIVE-LOAD    STRESSES 


103 


TABLE  11.— Continued 

MAXIMUM  BENDING  MOMENTS  IN  GIRDER  BRIDGES  WITHOUT  FLOOR-BEAMS, 
COOPER'S  #50  LOADING 

Values  in  Thousands  of  Foot-Pounds  per  Rail 
SHORTER  SEGMENT  h 


250 
225 
200 
175 
160 
150 
140 
130 
120 
110 
100 
95 
90 
85 
80 
75 
70 
65 

65 

70 

75 

80 

85 

90 

95 

100 

110 

120 

130 

140 

22909 
20799 
18674 
16530 
15231 
14359 
13488 
12610 
11725 
10832 
9954 
9552 
9129 
8679 
8228 
7746 
7237 
6718 

24594 
22327 
20045 
17756 
16371 
15443 
14510 
13571 
12625 
11672 
10709 
10227 
9771 
9285 
8804 
8286 
7746 

26327 
23904 
21465 
19009 
17523 
16492 
15494 
14492 
13482 
12465 
11438 
10921 
10401 
9896 
9375 
8821 

28026 
25439 
22836 
20214 
18633 
17573 
16508 
15436 
14357 
13270 
12173 
11620 
11064 
10505 
9943 

29707 
26961 
24200 
21417 
19736 
18609 
17475 
16336 
15091 
14033 
12867 
12280 
11690 
11095 

31355 
28446 
25522 
22521 
20795 
19601 
18402 
17195 
15984 
14765 
13536 
12917 
12295 

32955 
29885 
26800 
23690 
21812 
20553 
19288 
18016 
16776 
15490 
14185 
13543 

34575 
31347 
28102 
24835 
22861 
21539 
20211 
18876 
17533 
16182 
14821 

37690 
34144 
30581 
26996 
24832 
23382 
21927 
20465 
18996 
17518 

40739 
36878 
33000 
29098 
26745 
25189 
23588 
22000 
20406 

43791 
39614 
35414 
31204 
28662 
26961 
25254 
23542 

46819 
42327 
37819 
33289 
30558 
28732 
26900 



« 

or  1,  and  I,  each  >  142  ft.  M  =  1.25  1,  L  -f  4750  -f 

Li 

0    0 

O  C 

•)  O    O  O  O 

o 

'///////////////A 

<///////' 

i 

e  n 

'k 

1 

Influence  Line  for  M 

^ 

^Qr~^  —  —  -_ 

*t 

104 


LIVE-LOAD    STRESSES 


TABLE  12 

MAXIMUM  BENDING  MOMENTS  IN  GIRDER  BRIDGES  WITHOUT  FLOOR-BEAMS, 
COOPER'S  E6Q  LOADING 

Values  in  Thousands  of  Foot-pounds  per  Rail 
SHORTER  SEGMENT  h 


5 

10 

15 

20 

or 

30 

35 

40 

45 

50 

55 

60 

250 

2302 

4547 

6772 

8969 

11117 

13230 

15305 

17342 

19374 

21418 

23442 

25474 

225 

2106 

4153 

6184 

8183 

10136 

12052 

13932 

15773 

17615 

19464 

21298 

23134 

200 

1909 

3757 

5591 

7390 

9146 

10862 

12547 

14190 

15840 

17497 

19139 

20773 

175 

1709 

3354 

4990 

6584 

8144 

9658 

11146 

12587 

14046 

15509 

16958 

18400 

160 

1579 

3109 

4622 

6095 

7534 

8924 

10294 

11612 

12958 

14303 

15636 

16950 

150 

1505 

2944 

4375 

5765 

7123 

8430 

9720 

10956 

12224 

13492 

14749 

15996 

140 

1421 

2777 

4126 

5430 

6707 

7931 

9140 

10294 

11486 

12674 

13854 

15024 

130 

1337 

2608 

3872 

5090 

6287 

7427 

8555 

9625 

10741 

11851 

12953 

14045 

120 

1250 

2437 

3614 

4746 

5860 

6912 

7961 

8946 

9986 

11017 

12042 

13056 

110 

1162 

2263 

3352 

4394 

5425 

6390 

7357 

8270 

9222 

10174 

11122 

12058 

7* 

100 

1070 

2084 

3083 

4034 

4980 

5864 

6742 

7579 

8476 

9352 

10219 

11081 

"a 

95 

1024 

1993 

2945 

3850 

4753 

5596 

6436 

7296 

8147 

8987 

9820 

10644 

£ 

90 

974 

1896 

2800 

3666 

4524 

5324 

6172 

6991 

7802 

8602 

9395 

10178 

§? 

85 

925 

1800 

2656 

3472 

4282 

5047 

5885 

6662 

7404 

8188 

8938 

9673 

m 

80 

876 

1702 

2507 

3280 

4042 

4800 

5573 

6307 

7034 

7757 

8470 

9175 

fe 

75 

827 

1604 

2359 

3082 

3796 

4512 

5233 

5946 

6634 

7312 

7980 

8641 

fcJD 

70 

774 

1505 

2212 

2885 

3550 

4207 

4882 

5558 

6198 

6829 

7451 

8068 

o 

65 

722 

1397 

2051 

2688 

3304 

3903 

4529 

5155 

5747 

6331 

6912 

7489 

60 

679 

1296 

1898 

2473 

3037 

3583 

4156 

4732 

5279 

5826 

6365 

6895 

55 

637 

1207 

1758 

2276 

2784 

3293 

3818 

4326 

4820 

5270 

5789 

50 

595 

1124 

1637 

2096 

2569 

3035 

3504 

3952 

4392 

4829 

45 

551 

1038 

1507 

1936 

2351 

2771 

3204 

3606 

4003 

40 

503 

953 

1376 

1757 

2129 

2503 

2881 

3240 

35 

452 

856 

1229 

1574 

1908 

2234 

2561 

30 

406 

758 

1081 

1378 

1663 

1940 

25 

353 

660 

934 

1181 

1418 

20 

300 

559 

776 

984 

15 

224 

45C 

616 

10 

15C 

30C 

5 

74 

For  h  and  k  each  >  142  ft.  M  =  1.5 


+  5700^ 
LI 


LIVE-LOAD    STRESSES 


105 


TABLE  12.— Continued 

MAXIMUM  BENDING  MOMENTS  IN  GIRDER  BRIDGES  WITHOUT  FLOOR-BEAMS, 
COOPER'S  #60  LOADING 

Values  in  Thousands  of  Foot-pounds  per  Rail 
SHORTER  SEGMENT  h 


65 

70 

75 

80 

85 

90 

95    100 

110 

120 

130 

140 

250 
225 
200 
175 
160 
150 
40 
130 
120 
110 
100 
95 
90 
85 
80 
75 
70 
65 

27491 
24959 
22409 
19836 
18277 
17231 
16186 
15132 
14070 
12998 
11945 
11462 
10955 
10415 
9874 
9295 
8684 
8062 

29513 
26792 
24054 
21307 
19645 
18532 
17412 
16285 
15150 
14006 
12851 
12272 
11725 
11142 
10565 
9943 
9295 

31592 
28685 
25758 
22811 
21028 
19790 
18593 
17390 
16178 
14958 
13726 
13105 
12481 
11875 
11250 
10585 

33631 
30527 
27403 
24257 
22360 
21088 
19810 
18523 
17228 
15924 
14608 
13944 
13277 
12606 
11932 

35648 
32353 
29040 
25700 
23683 
22331 
20970 
19603 
18110 
16840 
15440 
14736 
14028 
13314 

37626 
34135 
30626 
27025 
24954 
23521 
22082 
20634 
19181 
17718 
16243 
15500 
14754 

39546 
35862 
32160 
28428 
26174 
24664 
23146 
21619 
20131 
18588 
17022 
16252 

41490 
37616 
33722 
29802 
27433 
25847 
24253 
22651 
21040 
19418 
17785 

45228 
40973 
36697 
32395 
29798 
28058 
26312 
24558 
22795 
21022 

48887 
44254 
39600 
34918 
32094 
30227 
28306 
26400 
24487 

52549 
47537 
42497 
37444 
34394 
32353 
30305 
28250 

56183 
50792 
45383 
39947 
36670 
34478 
32280 

1 

For  k  and  k  each  >142  ft.  M  =  1. 


+  5700^ 

Li 


OOOO  noon 


Influence  Line  for  M 


106 


LIVE-LOAD    STRESSES 


TABLE  13 

MAXIMUM  PIER  REACTIONS  BETWEEN  EQUAL  AND  UNEQUAL  SPANS,  COOPER'S 

#40  LOADING 

Values  in  Thousands  of  Pounds  per  Rail 
SHORTER  SEGMENT  h 


0 

5 

10 

15 

20 

25 

30 

35 

40 

45 

50 

56 

250 

314 

314 

315 

318 

3?!? 

3?,6 

329 

332 

336 

338 

342 

346 

225 

287 

287 

290 

294 

298 

301 

304 

306 

309 

312 

317 

321 

200  
175  
160  
150  
140  
130 

261 
234 
218 
207 
196 
185 

261 
234 
218 
207 
196 
185 

263 
236 
220 
210 
198 
187 

268 
241 
225 
214 
203 
192 

271 
244 
228 
218 
206 
196 

275 
248 
232 
222 
210 

278 
251 
236 
225 
214 
203 

281 
254 
238 
229 
218 
?,08 

284 
258 
242 
231 
220 
210 

287 
262 
246 
234 
224 
214 

292 
266 
250 
239 
229 
219 

296 
269 
254 
244 
234 
?94 

120  ... 

174 

174 

176 

181 

184 

189 

19?, 

196 

198 

?,04 

208 

9 
I 

110  
100  
95  
90  

162 
150 
144 
137 

162 
150 
144 
137 

165 
153 
146 
140 

170 
158 
151 
146 

173 
162 
155 
150 

178 
166 
160 
154 

181 
170 
163 
158 

185 
174 
168 
163 

188 
177 
173 
168 

193 
182 

178 
174 

198 

187 
182 
178 

202 
192 

188 
183 

85  
80  ... 

131 
124 

131 
124 

134 
127 

139 
133 

142 
137 

148 
142 

152 
146 

158 
153 

163 
158 

168 
163 

174 

168 

178 
174 

1 

75  
70  
65  

118 
110 
104 

118 
110 
104 

122 
114 
107 

126 
120 
112 

130 
124 
118 

135 

128 
122 

140 
134 
126 

146 
139 
133 

152 
146 
139 

158 
150 
144 

162 
156 
149 

167 
162 
155 

60 

98 

98 

101 

106 

110 

115 

119 

125 

131 

137 

142 

148 

55  
50  
•45  ...  . 

93 

87 

82 

93 

87 
82 

95 
90 

85 

99 
94 
90 

103 
98 
93 

108 
102 

98 

113 
108 
102 

118 
114 
109 

125 
118 
114 

130 
124 
118 

134 
129 

141 

40  

75 

75 

79 

84 

88 

92 

98 

102 

108 

35  

69 

69 

74 

78 

82 

87 

92 

98 

30  

63 

63 

67 

72 

77 

82 

86 

25  

57 

57 

6? 

66 

71 

76 

20 

50 

50 

56 

60 

66 

15 

40 

40 

50 

55 

10  

30 

30 

40 

5.. 

20 

20 

For  h  and  k  each  >  142  f t.  R  =  L  + 


3800 


LIVE-LOAD    STRESSES 


107 


TABLE  13— Continued 

MAXIMUM  PIER  REACTIONS  BETWEEN  EQUAL  AND  UNEQUAL  SPANS,  COOPER'S 

#40  LOADING 

Values  in  Thousands  of  Pounds  per  Rail 
SHORTER  SEGMENT  h 


60 

65 

70 

75 

80 

85 

90 

95 

100 

110 

120 

130 

140 

~s 

I 

-r 

250 
225 
200 
175 
160 
150 
140 
130 
1?0 

350 
326 
300 

274 
258 
248 
238 
229 
218 

356 
330 
305 
279 
264 
254 
242 
233 
222 

359 
334 
309 
284 
269 
259 
249 
239 
228 

365 
340 
314 
290 
274 
264 
253 
243 
233 

370 
345 
320 
294 
280 
269 
259 
250 
239 

374 
350 
324 
300 

284 
274 
264 
254 
242 

379 
354 
329 
303 

289 
278 
270 
258 
248 

382 
358 
333 
308 
293 
282 
273 
262 
253 

387 
362 
337 
312 

297 

287 
277 
267 
?57 

395 
370 
345 
319 
305 
295 
284 
274 
265 

402 
377 
352 
327 
312 
302 
292 
282 
?,7?, 

410 
385 
359 
334 
320 
310 
299 
290 

417 
392 
367 
342 
328 
318 
308 

1 

110 
100 
95 
PO 

207 
197 
192 

188 

212 
202 
198 
194 

218 
208 
203 
198 

223 
214 
208 
203 

230 
219 
214 
209 

234 
224 
219 
214 

238 
229 
223 

218 

243 
233 
229 

247 
238 

255 

.  .  . 

85 

183 

189 

194 

198 

204 

209 

80 

178 

184 

188 

194 

199 

75 

173 

178 

183 

188 

70 

166 

171 

178 

65 

160 

165 

60 

153 

For  Zi  and  k  each  >142  ft.  R  =  L  +     p 


o  O  O  O  O    O  O  O 


108 


LIVE-LOAD    STRESSES 


TABLE  14 

MAXIMUM  PIER  REACTIONS  BETWEEN  EQUAL  AND  UNEQUAL  SPANS,  COOPER'S 

#50  LOADING 

Values  in  Thousands  of  Pounds  per  Rail 
SHORTER  SEGMENT  h 


0 

5 

10 

15 

20 

25 

30 

35 

40 

45 

50 

55 

250  
225  
200  
175  
160  
150  

392 
359 
326 
293 
273 
259 

392 
359 
326 
293 
273 
259 

394 
362 
329 
295 
275 
262 

398 
367 
335 
301 
281 
267 

403 
372 
339 
305 

285 
272 

407 
376 
344 
310 
290 
277 

411 

380 
347 
314 
295 

281 

415 
383 
351 
318 

298 
286 

420 
386 
355 
323 
302 
289 

423 
390 
359 
327 
307 
293 

428 
396 
365 
332 
313 
299 

432 
401 
370 
336 
318 
305 

140  

245 

245 

248 

254 

258 

263 

268 

273 

275 

280 

286 

293 

130 

231 

231 

234 

240 

245 

251 

?M 

260 

262 

268 

274 

280 

~$ 

1 

120  
110  
100  
95.  
90  
85  
80  

217 
202 
187 
180 
171 
164 
755 

217 
202 
187 
180 
171 
164 
155 

220 
206 
191 
183 
175 
168 
159 

226 
212 
197 
189 

182 
174 
166 

230 
216 
202 
194 

187 
178 
171 

236 
222 
208 
200 
192 
185 
177 

240 
226 
212 
204 
197 
190 
183 

245 
231 
218 
210 
204 
198 
191 

248 
235 
221 
216 
210 
204 
197 

255 
241 
227 
222 
218 
210 
204 

260 
247 
234 
228 
223 
217 
210 

266 
253 
240 
235 
229 
223 
217 

^ 

75 

147 

147 

152 

158 

163 

169 

175 

183 

190 

197 

203 

209 

I 

70  
65  
60  
55  
50  
45  
40 

138 
130 
123 
116 
109 
102 
94 

138 
130 
123 
116 
109 
102 
94 

143 
134 
126 
119 
112 
106 
99 

150 
140 
132 
124 
118 
112 
105 

155 
147 
137 
129 
122 
116 
110 

160 
152 
144 
135 
128 
122 
115 

167 
158: 
149 
141 
135 
128 
1?,?, 

174 
166 
156 
148 
142 
136 
128 

182 
174 
164 
156 
148 
142 
135 

188 
180 
171 
162 
155 
148 

195 
186 
178 
168 
161 

202 
194 
185 
176 

35  
30 

86 
79 

86 
79 

92 

84 

98 
90 

103 
96 

109 
102 

115 

108 

122 

25 

71 

71 

77 

83 

89 

95 

20 

63 

63 

70 

75 

82 

15 

50 

50 

62 

69 

10 

QQ 

QQ 

50 

5 

25 

25 

.  . 

For  li  and  k  each  >142  ft.  R  =  1.25  L  + 


4750 


LIVE-LOAD    STRESSES 


109 


TABLE  14.— Continued 

MAXIMUM  PIER  REACTIONS  BETWEEN  EQUAL  AND  UNEQUAL  SPANS,  COOPER'S 

#50  LOADING 

Values  in  Thousands  of  Pounds  per  Rail 
SHORTER  SEGMENT  l\ 


60 

65 

70 

75 

80 

85 

90 

95 

100 

110 

120 

130 

140 

250... 

437 

445 

449 

456 

463 

468 

474 

478 

484 

494 

502 

512 

521 

225... 

407 

413 

418 

425 

431 

437 

442 

448 

452 

462 

471 

481 

490 

200... 

375 

381 

386 

393 

400 

405 

411 

416 

421 

431 

440 

449 

459 

175... 

343 

349 

355 

362 

368 

375 

379 

385 

390 

399 

409 

418 

427 

160... 

323 

330 

336 

343 

350 

355 

361 

366 

371 

381 

390 

400 

410 

150... 

310 

317 

324 

330 

336 

343 

348 

353 

359 

369 

378 

387 

397 

140... 

298 

303 

311 

316 

324 

330 

337 

341 

346 

355 

•365 

374 

385 

130... 

286 

291 

299 

304 

312 

317 

323 

328 

334 

343 

352 

362 

120... 

272 

278 

285 

291 

299 

303 

310 

316 

321 

331 

340 

110... 

259 

265 

273 

279 

287 

292 

298 

304 

309 

319 

100... 

246 

253 

260 

267 

274 

280 

286 

291 

296 

95... 

240 

247 

254 

260 

267 

274 

279 

286 

90. 

235 

242 

248 

254 

261 

268 

273 

85... 

229 

236 

242 

248 

255 

261 

80... 

223 

230 

235 

242 

249 

75... 

216 

222 

229 

235 

70... 

208 

214 

222 

65.  .  . 

200 

206 

60... 

191 

For  Zi  and  k  each  >  142  ft.  R  =  1.25  L  -f 


4750 


O   O  O  O  O      O    O    O   O 


110 


LIVE-LOAD    STRESSES 


TABLE  15 

MAXIMUM  PIER  REACTIONS  BETWEEN  EQUAL  AND  UNEQUAL  SPANS,  COOPER'S 

#60  LOADING 

Values  in  Thousands  of  Pounds  per  Rail 
SHORTER  SEGMENT  h 


0 

5 

10 

15 

20 

25 

30 

35 

40 

45 

50 

55 

250.  . 

470 

470 

473 

478 

484 

488 

493 

498 

504 

508 

514 

518 

225  

431 

431 

434 

440 

446 

451 

456 

460 

463 

468 

475 

481 

200  
175 

391 
352 

391 
352 

395 
354 

402 
361 

407 
366 

413 
372 

417 

377 

421 

38? 

426 

388 

431 
39? 

438 
398 

444 
403 

160  . 

328 

328 

330 

337 

342 

348 

354 

358 

36? 

368 

376 

38? 

150  
140  
130  

311 
294 
?77 

311 
294 

?,77 

314 
298 
?,81 

320 
305 

288 

326 
310 
294 

332 
316 
301 

337 
322 
305 

343 

328 
31? 

347 
330 
314 

352 
336 
3?,?, 

359 
343 
3?,9 

366 
352 
336 

120  

?60 

?60 

?64 

?71 

?76 

283 

?88 

?94 

?98 

306 

31?, 

319 

Segment  Z2 

110  
100  
95  
90  
85  
80  
75  

242 
224 
216 
205 
197 
186 
176 

242 
224 
216 
205 
197 
186 
176 

247 
229 
220 
210 
202 
191 
18'? 

254 
236 
227 
218 
209 
199 
190 

259 
242 
233 
224 
214 
205 
196 

266 
250 
240 
230 
222 
212 
203 

271 
254 
245 
236 
228 
220 
210 

277 
262 
252 
245 
238 
229 
220 

282 
265 
259 
252 
245 
236 

??8 

289 
272 
266 
262 
252 
245 
?,36 

296 
281 
274 
268 
260 
252 
?44 

304 
288 
282 
275 
268 
260 
?51 

1 

70  
65 

166 
156 

166 
156 

172 
161 

180 
168 

186 
176 

192 
182 

200 
190 

209 
199 

218 
?09 

226 
?16 

234 
??3 

242 
?33 

9 

60 

148 

148 

151 

158 

164 

173 

179 

187 

197 

?05 

?14 

??? 

H-l 

55  
50  
45 

139 
131 
122 

139 
131 
122 

143 
134 
127 

149 
142 
134 

155 
146 
139 

162 
154 
146 

169 
162 
154 

178 
170 
163 

187 
178 
170 

194 
186 

178 

202 
193 

211 

40 

113 

113 

119 

126 

132 

138 

146 

154 

16? 

35 

103 

103 

110 

118 

124 

131 

138 

146 

30  
25 

95 

85 

95 

85 

101 
92 

108 
100 

115 
107 

122 
114 

130 

20 

76 

76 

84 

90 

98 

15 

60 

60 

74 

83 

10.. 
5 

46 
30 

46 
30 

60 

For  Zj  and  Z2  each  >142  ft.  R 


1    Z    T      i      57GO 

1.5  L  -f  -- 


LIVE-LOAD    STRESSES 


111 


TABLE  15.— Continued 

MAXIMUM  PIER  REACTIONS  BETWEEN  EQUAL  AND  UNEQUAL  SPANS,  COOPER'S 

#60  LOADING 

Values  in  Thousands  of  Pounds  per  Rail 
SHORTER  SEGMENT  Zi 


60 

65 

70 

75 

80 

85 

90 

95 

100 

110 

120 

130 

140 

250... 
225... 
200... 
175... 
160... 
150... 
140... 
130... 
120 

524 
488 
450 
412 
388 
372 
358 
343 
326 

534 
496 
457 
419 
396 
380 
364 
349 
334 

539 
502 
463 
426 
403 
389 
373 
359 
342 

547 
510 
472 
434 
412 
396 
379 
365 
349 

556 
517 

480 
442 
420 
403 
389 
374 
359 

562 
524 
486 
450 
426 
412 
396 
380 
364 

569 
530 
493 
455 
433 
418 
404 
388 
372 

574 

538 
499 
462 
439 
424 
409 
394 
379 

581 
542 
505 
468 
445 
431 
415 
401 
385 

593 
554 
517 
479 
457 
443 
426 
412 
397 

602 
565 

528 
491 
468 
454 
438 
422 
408 

614 
577 
539 
502 
480 
464 
449 
434 

625 
588 
551 
512 
492 
476 
462 

110... 
100... 
95 

311 
295 

288 

318 
304 
296 

328 
312 
305 

335 
320 
312 

344 
329 
320 

350 
336 
329 

358 
343 
335 

365 
349 
343 

371 
356 

383 

90 

282 

290 

298 

305 

313 

322 

328 

85 

275 

283 

290 

298 

306 

313 

80 

268 

276 

282 

290 

299 

75. 

259 

266 

275 

282 

70... 
65... 
60.. 

250 
240 
229 

257 

247 

266 

.  .  . 

... 

For  h  and  Z2  each  >142  ft.  R  =  1.5  L  + 


5700 


112 


LIVE-LOAD    STRESSES 


TABLE  16 
EQUIVALENT  UNIFORM  LOADS  FOR  COOPER'S  #40  LOADING 

Values  in  Pounds  per  Lineal  Foot  per  Rail 
SHORTER  SEGMENT  l\ 


Longer  Segment  lz 

o 

5 

10 

15 

20 

25 

30 

35 

40 

45 

50 

55 

2270 
2300 
2320 
2340 
2370 
2380 
2400 
2420 
2430 
2460 
2480 
2500 
2540 
2550 
2570 
2580 
2580 
2580 
2580 
2550 

250 

2500 
2550 
2610 
2680 
2730 
2760 
2800 
2850 
2900 
2940 
3000 
3020 
3050 
3080 
3110 
3140 
3160 
3190 
3270 
3370 
3490 
3630 
3770 
3960 
4200 
4540 
5000 
5336 
6000 
8000 

2450 
2500 
2540 
2610 
2630 
2670 
2700 
2740 
2770 
2810 
2850 
2880 
2890 
2920 
2920 
2940 
2940 
2960 
3020 
3090 
3180 
3260 
3350 
3450 
3610 
3770 
4000 
4000 
4000 
4000 

2430 
2460 
2500 
2550 
2590 
2620 
2650 
2670 
2710 
2740 
2780 
2800 
2810 
2820 
2840 
2860 
2870 
2870 
2880 
2930 
3000 
3080 
3180 
3260 
3380 
3520 
3730 
4000 
4000 

2410 
2450 
2490 
2540 
2570 
2590 
2620 
2650 
2680 
2710 
2740 
2760 
2770 
2780 
2790 
2800 
2810 
2810 
2820 
2840 
2910 
2980 
3060 
3120 
3200 
3320 
3450 
3650 

2380 
2430 
2460 
2510 
2540 
2570 
2580 
2610 
2640 
2660 
2690 
2700 
2720 
2730 
2740 
2740 
2750 
2760 
2750 
2760 
2800 
2870 
2930 
3010 
3060 
3150 
3280 

2370 
2400 
2440 
2490 
2510 
2540 
2560 
2580 
2610 
2630 
2660 
2670 
2680 
2700 
2710 
2700 
2700 
2700 
2700 
2700 
2740 
2780 
2840 
2900 
2960 
3020 

2350 
2380 
2420 
2460 
2480 
2500 
2520 
2540 
2560 
2580 
2610 
2620 
2630 
2640 
2670 
2670 
2670 
2670 
2660 
2660 
2700 
2740 
2780 
2840 
2880 

2330 
2360 
2390 
2420 
2450 
2460 
2490 
2510 
2530 
2550 
2570 
2580 
2620 
2640 
2660 
2660 
2660 
2660 
2640 
2650 
2670 
2710 
2740 
2790 

2310 
2340 
2370 
2400 
2420 
2430 
2450 
2470 
2490 
2500 
2530 
2560 
2590 
2620 
2620 
2640 
2650 
2650 
2630 
2620 
2630 
2670 
2700 

2300 
2320 
2350 
2380 
2400 
2420 
2430 
2450 
2460 
2490 
2510 
2540 
2570 
2580 
2610 
2620 
2620 
2620 
2610 
2600 
2600 
2640 

2290 
2310 
2340 
2360 
2380 
2400 
2420 
2430 
2450 
2460 
2500 
2520 
2550 
2570 
2580 
2600 
2600 
2600 
2590 
2560 
2580 

225  .  ... 

200 

175  

160  
150  

140 

130  
120  
110  . 

100 

95  
90  
85  

80 

75 

70  
65  .  . 

60  
55  
50  
45  
40  
35  
30  
25  
20.  . 

15  
10  
5  

. 

i 

For  h  and  12  each  >  142  ft.  q 


1000 


LIVE-LOAD    STRESSES 


113 


TABLE  16.— Continued 
EQUIVALENT  UNIFORM  LOADS  FOR  COOPER'S  .#40  LOADING 

Values  in  Pounds  per  Lineal  Foot  per  Rail 
SHORTER  SEGMENT  h 


60 

65 

70 

75 

80   85 

90 

95 

100 

110 

120 

130 

140 

250.  . 
225  
200  
175  
160  
150  
140  
130  

2260 
2290 
2310 
2340 
2350 
2370 
2380 
2400 
2420 
2440 
2460 
2500 
2510 
2530 
2550 
2560 
2560 
2560 
2550 

2260 
2280 
2300 
2320 
2340 
2350 
2380 
2390 
2410 
2420 
2460 
2480 
2500 
2510 
2540 
2540 
2540 
2540 

2250 
2270 
2290 
2320 
2340 
2360 
2370 
2390 
2410 
2420 
2450 
2460 
2480 
2500 
2520 
2530 
2530 

2250 
2270 
2290 
2320 
2340 
2350 
2360 
2380 
2400 
2420 
2440 
2460 
2460 
2490 
2500 
2510 

2240 
2260 
2280 
2310 
2330 
2340 
2360 
2380 
2400 
2420 
2440 
2450 
2460 
2470 
2490 

2230 
2260 
2280 
2300 
2320 
2340 
2350 
2370 
2370 
2400 
2420 
2440 
2450 
2460 

2220 
2250 
2270 
2290 
2310 
2330 
2340 
2350 
2370 
2390 
2410 
2420 
2430 

2220 
2240 
2260 
2280 
2300 
2300 
2320 
2340 
2350 
2380 
2390 
2400 

2210 
2220 
2250 
2270 
2280 
2300 
2310 
2330 
2340 
2350 
2380 

2200 
2220 
2230 
2240 
2260 
2270 
2280 
2290 
2300 
2320 

2180 
2180 
2200 
2210 
2230 
2240 
2250 
2260 
2270 

2160 
2170 
2180 
2200 
2210 
2220 
2220 
2230 

2140 
2150 
2160 
2180 
2180 
2190 
2200 

120  
110  
100  ... 



95  
90  
85  
80  
75  
70  
65 

60.  . 

For  ?!  and  12  each   >  142  ft.  q 


(2.0  +  ^1000 


liL  I 


Pounds  nor  Lineal  Foot  per  Rail. 


114 


LIVE-LOAD    STRESSES 


TABLE  17 
EQUIVALENT  UNIFORM  LOADS  FOR  COOPER'S  #50  LOADING 

Values  in  Pounds  per  Lineal  Foot  per  Rail 
SHORTER  SEGMENT  h 


-s 

1 

J 

0 

5 

10 

15 

20 

25 

30 

35 

40 

45 

50 

55 

2840 
2870 
2900 
2930 
2960 
2980 
3000 
3020 
3040 
3065 
3095 
3130 
3165 
3185 
3210 
3225 
3225 
3220 
3215 
3190 

250.. 
225  

3130 
3190 
3265 
3350 
3410 
3455 
3505 
3560 
3620 
3680 
3750 
3780 
3810 
3850 
3885 
3920 
3945 
3990 
4085 
4215 
4360 
4540 
4715 
4945 
5255 
5680 
6250 
6670 
7500 
10000 

3060 
3120 
3180 
3260 
3290 
3340 
3380 
3420 
3460 
3510 
3560 
3600 
3610 
3650 
3650 
3670 
3680 
3700 
3780 
3860 
3970 
4080 
4190 
4310 
4510 
4710 
5000 
5000 
5000 
5000 

3040 
3080 
3130 
3190 
3240 
3270 
3305 
3340 
3385 
3430 
3470 
3500 
3510 
3530 
3545 
3565 
3585 
3580 
3595 
3660 
3750 
3850 
3975 
4080 
4215 
4400 
4660 
5000 
5000 

3010 
3060 
3110 
3170 
3210 
3240 
3275 
3310 
3350 
3385 
3425 
3445 
3455 
3470 
3480 
3495 
3510 
3505 
3515 
3550 
3635 
3720 
3825 
3900 
4000 
4150 
4315 
4560 

2980 
3040 
3080 
3140 
3170 
3210 
3230 
3260 
3295 
3330 
3360 
3375 
3395 
3405 
3415 
3425 
3435 
3445 
3435 
3450 
3495 
3585 
3660 
3760 
3825 
3935 
4100 

2960 
3000 
3050 
3110 
3140 
3170 
3195 
3225 
3255 
3285 
3320 
3340 
3350 
3370 
3385 
3380 
3380 
3375 
3375 
3380 
3425 
3480 
3550 
3630 
3695 
3780 

2940 
2980 
3020 
3070 
3100 
3130 
3150 
3175 
3200 
3225 
3260 
3275 
3290 
3300 
3335 
3340 
3340 
3335 
3315 
3325 
3370 
3420 
3475 
3545 
3595 

2910 
2950 
2990 
3030 
3060 
3080 
3110 
3135 
3160 
3185 
3210 
3225 
3265 
3295 
3315 
3325 
3320 
3325 
3300 
3305 
3335 
3390 
3430 
3485 

2890 
2920 
2960 
3000 
3020 
3040 
3064 
3085 
3106 
3133 
3158 
3200 
3237 
3266 
3284 
3303 
3308 
3305 
3286 
3277 
3293 
3339 
3375 

2870 
2900 
2940 
2970 
3000 
3020 
3040 
3060 
3080 
3105 
3140 
3175 
3210 
3225 
3255 
3275 
3280 
3270 
3260 
3245 
3250 
3295 

2860 
2890 
2920 
2950 
2980 
3000 
3018 
3039 
3060 
3083 
3117 
3153 
3186 
3210 
3232 
3250 
3252 
3246 
3237 
3194 
3219 

200  

175  . 

160.  .  .  . 
150  
140  
130  
120  
110  
100  
95.... 
90.  ... 
85  
80  
75  
70  
65  
60.. 
55.... 
50  
45  
40  
35  
30  



25.. 

20...... 

15  
10  
5  

For  h  and  k  each  >142  ft. 


LIVE-LOAD    STRESSES 


115 


TABLE  17.— Continued 
EQUIVALENT  UNIFORM  LOADS  FOR  COOPER'S  E5Q  LOADING 

Values  in  Pounds  per  Lineal  Foot  per  Rail 
SHORTER  SEGMENT  h 


60    I    65 


70 


75        80 


250 2830  2820  2810J2810  2800J2790  2780  2770  2760J2750  2720  2700  2680 

225 2860  2850  284012840  2830  2820  281012800!27802770i2730>2710'2690 

200 S2890  287012860  2860  2850  2850  2840 ;  2820 !  28 10  2790  2750 12720 12700 

2920  2900:2900  2900i2890j2880  2860  2850,2840  2800|2760  2750! 2720 
2940i2930|2920  2920  2910  2900  2890  2870|2850  2820  2790  2760J2730 
2960  2940|2950  2940  2930  2920  2910|2880  2870  2840  2800  27701 2740 
2980  2965  2960  2950i2950  2940  2920  2900  2890  2850|2810  2775  2750 
3000  2985  2985  2975'2970  2955  2940  2920 \ 2905  2860  2820i2785 
3020  3005  3005  2995s  2995  2960  2960  2940  2920  2880  2835 
304513030  3030  3020^3015  3000  2985  2965  2940  2895 
3080  3065  3060  3050| 3045 1 3030  301012985  2965 
3115  3095  3075  3065^3060  3050  3020 !  3001 
3140  3120  3100  3080  3075  3060  3035 


175 

160 

150 

140 

130 

120 

110 

100 

95 

90 

85 

80 

75 

70 

65 

60.  . 


90 


95       100      110      120      130 


140 


3160  3140  3120  31053090  3070 

31853165314531253110 

3200318031553140 

320031803160 

3200:3180 

3190  .  . 


For  h  and  I,  each  >142  ft.  q  =  /2.5  +-f 

\     h 


1000 


Pounds  per  Lineal  Foot  per  Rail. 


116 


LIVE-LOAD    STRESSES 


TABLE  18 
EQUIVALENT  UNIFORM  LOADS  FOR  COOPER'S  #60  LOADING 

Values  in  Pounds  per  Lineal  Foot  per  Rail 
SHORTER  SEGMENT  h 


Longer  Segment  k 

0 

5 

10 

15 

20 

25 

30 

35 

40 

45 

•;o 

55 

250.. 
225  
200  
175  
160  
150  
140  :.... 
130 

3760 
3830 
3920 
4020 
4090 
4150 
4210 
4270 
4340 
4420 
4500 
4540 
4570 
4620 
4660 
4700 
4730 
4790 
4900 
5060 
5230 
5450 
5660 
5930 
6310 
6820 
7500 
8000 
9000 
12000 

3670 
3740 
3820 
3910 
3950 
4010 
4060 
4110 
4150 
4210 
4270 
4320 
4330 
4380 
4380 
4400 
4420 
4440 
4540 
4630 
4760 
4900 
5030 
5-170 
5410 
5650 
6000 
6000 
6000 
6000 

3650 
3700 
3760 
3830 
3890 
3920 
3970 
4010 
4070 
4120 
4160 
4200 
4210 
4240 
4260 
4280 
4310 
4300 
4320 
4390 
4500 
4620 
4780 
4900 
5060 
5280 
5590 
6000 
6000 

3610 
3670 
3730 
3800 
3850 
3890 
3940 
3970 
4020 
4070 
4120 
4140 
4150 
4160 
4180 
4200 
4210 
4210 
4220 
4260 
4370 
4460 
4600 
4680 
4800 
4980 
5180 
5470 

3580 
3650 
3700 
3770 
3800 
3850 
3880 
3910 
3960 
4000 
4030 
4060 
4080 
4080 
4100 
4120 
4130 
4140 
4130 
4140 
4200 
4310 
4390 
4510 
4600 
4730 
4920 

3550 
3600 
3660 
3730 
3770 
3800 
3840 
3850 
3910 
3950 
3980 
4010 
4020 
4040 
4070 
4060 
4060 
4060 
4060 
4060 
4120 
4180 
4260 
4360 
4440 
4540 

3530 
3580 
3620 
3680 
3720 
3760 
3780 
3820 
3840 
3880 
3910 
3940 
3950 
3960 
4010 
4010 
4010 
4010 
3980 
4000 
4040 
4100 
4180 
4260 
4320 

3490 
3540 
3590 
3640 
3670 
3700 
3730 
3770 
3790 
3830 
3850 
3880 
3920 
3960 
3980 
4000 
3980 
4000 
3960 
3970 
4010 
4070 
4120 
4190 

3470 
3500 
3550 
3600 
3620 
3650 
3680 
3710 
3730 
3760 
3790 
3840 
3890 
3920 
3940 
3960 
3970 
3970 
3950 
3940 
3950 
4010 
4060 

3440 
3480 
3530 
3560 
3600 
3620 
3650 
3670 
3700 
3760 
3770 
3820 
3850 
3880 
3910 
3940 
3940 
3920 
3910 
3900 
3900 
3960 

3430 
3470 
3500 
3540 
3580 
3600 
3630 
3650 
3670 
3700 
3740 
3780 
3830 
3850 
3880 
3900 
3900 
3900 
3890 
3840 
3860 

3410 
3440 
3480 
3520 
3550 
3580 
3600 
3620 
3650 
3680 
3720 
3760 
3800 
3830 
3850 
3870 
3870 
3860 
3860 
3830 

120  
110  
100  
95  
90  
85  
80  
75  
70  
65  
60  
55  
50  
45  
40  
35  
30  
25  

20 

15  
10  
5  

For  J,  and  k  each  >142  ft.  q  =  (s.O  +  -}~ 

*  liL 


1000 


LIVE-LOAD    STRESSES 


117 


TABLE  18.— Continued 
EQUIVALENT  UNIFORM  LOADS  FOR  COOPER'S  #60  LOADING 

Values  in  Pounds  per  Lineal  Foot  per  Rail 
SHORTER  SEGMENT  h 


60 

65 

70 

75 

80 

85 

90 

95 

100 

110 

120 

130 

140 

250  

3400 
3430 
3470 
3500 
3530 
3550 
3580 
3600 
3620 
3650 
3700 
3740 
3770 
3790 
3830 
3840 
3840 
3840 
3830 

3380 
3420 
3440 
3480 
3520 
3530 
3560 
3590 
3610 
3640 
3680 
3720 
3740 
3770 
3800 
3820 
3820 
3820 

3370 
3410 
3430 
3480 
3500 
3540 
3550 
3580 
3600 
3640 
3670 
3690 
3720 
3740 
3770 
3780 
3790 

3370 
3410 
3430 
3480 
3500 
3530 
3540 
3570 
3590 
3630 
3660 
3680 
3700 
3730 
3750 
3770 

3360 
3400 
3420 
3470 
3490 
3520 
3540 
3560 
3590 
3620 
3650 
3670 
3690 
3710 
3730 

3350 
3380 
3420 
3460 
3480 
3500 
3530 
3550 
3550 
3600 
3640 
3660 
3670 
3680 

3340 
3370 
3410 
3430 
3470 
3490 
3530 
3550 
3550 
3590 
3610 
3620 
3650 

3320 
3360 
3380 
3420 
3440 
3460 
3480 
3500 
3530 
3560 
3590 
3600 

3310 
3340 
3370 
3410 
3420 
3440 
3470 
3490 
3500 
3530 
3560 

3300 
3320 
3350 
3360 
3380 
3410 
3420 
3430 
3460 
3480 

3260 
3280 
3300 
3310 
3350 
3360 
3370 
3380 
3410 

3240 
3250 
3260 
3300 
3310 
3320 
3340 
3350 

3220 
3230 
3240 
3260 
3280 
3290 
3300 

225  

200  
175  
160  
150  
140  
130  
120  
110  
100  
95  
90  
85  
80 

75 

70  
65  
60  

For  ?!  and  I*  each  >  142  ft.  q  =  (s.O  + 


1000 


Pounds  per  "Lineal  Foot  per  Rail 


118 


LIVE-LOAD    STRESSES 


TABLE  19 

INFLUENCE-LINE  ORDINATES  FOR  M  FOR  GIRDER  BRIDGES  WITHOUT  FLOOR- 
BEAMS 

Values  of  ^ 
SHORTER  SEGMENT  h 


Longer  begment  Iz 

5 

10 

15 

20 

25 

30 

35 

40 

45 

50 

55 

60 

250.. 
225   .  . 

4.90 
4.90 
4.88 
4.85 
4.85 
4.83 
4.83 
4.81 
4.80 
4.78 
4.76 
4.75 
4.74 
4.72 
4.71 
4.69 
4.67 
4.64 
4.62 
4.58 
4.55 
4.50 
4.44 
4.37 
4.29 
4.17 
4.00 
3.75 
3.33 
2.50 

9.62 
9.62 
9.52 
9.43 
9.43 
9.35 
9.34 
9.29 
9.23 
9.17 
9.09 
9.05 
9.00 
8.94 
8.89 
8.83 
8.75 
8.67 
8.58 
8.46 
8.33 
8.18 
8.00 
7.78 
7.50 
7.14 
6.67 
6.00 
5.00 

14.14 
14.06 
13.97 
13.83 
13.70 
13.64 
13.55 
13.44 
13.33 
13.19 
13.05 
12.95 
12.85 
12.76 
12.63 
12.50 
12.35 
12.20 
12.00 
11.79 
11.53 
11.25 
10.91 
10.50 
10.00 
9.38 
8.58 
7.50 

18.5 
18.4 
18.2 
17.9 
17.8 
17.6 
17.5 
17.3 
17.2 
16.9 
16.7 
16.5 
16.4 
16.2 
16.0 
15.8 
15.6 
15.3 
15.0 
14.7 
14.3 
13.9 
13.3 
12.7 
12.0 
11.1 
10.0 

22.7 
22.5 
22.2 
21.9 
21.6 
21.5 
21.2 
21.0 
20.7 
20.4 
20.0 
19.8 
19.6 
19.3 
19.0 
18.8 
18.4 
18.0 
17.6 
17.2 
16.7 
16.1 
15.4 
14.6 
13.6 
12.5 

26.7 
26.5 
26.1 
25.6 
25.3 
25.0 
24.7 
24.4 
24.0 
23.6 
23.1 
22.8 
22.5 
22.2 
21.8 
21.5 
21.0 
20.5 
20.0 
19.4 
18.8 
18.0 
17.2 
16.2 
15.0 

30.7 
30.3 
20.9 
29.2 
28.7 
28.4 
28.0 
27.6 
27.1 
26.6 
25.9 
25.6 
25.2 
24.8 
24.3 
23.9 
23.4 
22.7 
22.1 
21.4 
20.6 
19.7 
18.7 
17.5 

34.5 
33.9 
33.3 
32.6 
32.0 
31.6 
31.1 
30.6 
30.0 
29.3 
28.6 
28.1 
27.7 
27.2 
26.7 
26.1 
25.5 
24.8 
24.0 
23.2 
22.2 
21.2 
20.0 

38.2 
37.6 
36.8 
35.8 
35.2 
34.7 
34.1 
33.4 
32.7 
31.9 
31.1 
30.6 
30.0 
29.4 
28.8 
28.1 
27.4 
26.6 
25.8 
24.8 
23.7 
22.5 

41.7 
41.0 
40.0 
38.9 
38.0 
37.6 
36.8 
36.1 
35.3 
34.4 
33.3 
32.8 
32.2 
31.5 
30.8 
30.0 
29.2 
28.3 
27.3 
26.2 
25.0 

45.2 
44.2 
43.1 
42.0 
41.0 
40.3 
39.5 
38.6 
37.7 
36.6 
35.5 
34.8 
34.1 
33.4 
32.6 
31.8 
30.8 
29.8 
28.7 
27.5 

48.3 
47.4 
46.1 
44.6 
43.7 
42.9 
42.0 
41.0 
40.0 
38.7 
37.5 
36.7 
36.1 
35.2 
34.3 
33.3 
32.4 
31.2 
30.0 

200  
175  

160  

150  

140  
130  
120  
110  

100  

95  
90  

85 

80 

75  
70  
65  
60  
55  
50  
45  
40  





35  
30  . 

25  
20  
15  
10  
5...... 





LIVE-LOAD    STRESSES 


119 


TABLE  19.— Continued 

INFLUENCE-LINE  ORDINATES  FOR  M  FOR  GIRDER  BRIDGES  WITHOUT  FLOOR- 
BEAMS 

Values  of  ^-2 
LI 

SHORTER  SEGMENT  h 


65 

70 

75 

80 

85 

90 

95 

100 

110 

120 

130 

140 

250   . 

51.5 
50.5 
49.0 
47.2 
46.1 
45.2 
44.4 
43.3 
42.2 
40.8 
39.4 
38.6 
37.7 
36.8 
35.8 
34.8 
33  8 

54.6 
53.2 
51.8 
50.0 
48.5 
47.6 
46.7 
45.5 
44.3 
42.7 
41.2 
40.3 
39.4 
38.3 
137.3 
i36.2 
35  0 

57.5 
56.2 
54.6 
52.4 
51.0 
50.0 
49.0 
47.6 
46.3 
44.6 
42.9 
42.0 
41.0 
39.8 
38.7 
37.5 

60.6 
58.8 
57.1 
54.9 
53.2 
52.1 
51.0 
49.5 
48.1 
46.3 
44.4 
43.5 
42.4 
41.2 
40.0 

63.3 
61.7 
59.5 
57.1 
55.6 
54.3 
52.9 
51.6 
49.8 
48.1 
46.1 
44.8 
43.7 
42.5 

66.2 
64.1 
62.1 
59.5 
57.5 
56.2 
54.6 
53.2 
51.5 
49.5 
47.4 
46.3 
45.0 
.... 

69.0 
66.7 
64.5 
61.7 
59.5 
58.1 
56.5 
55.0 
53.2 
51.0 
48.8 
47.5 

71.4 
69.4 
66.8 
63.7 
61.7 
59.9 
58.5 
56.5 
54.6 
52.4 
50.0 

76.3 
73.5 
70.9 
67.6 
64.9 
63.3 
61.7 
59.5 
57.5 
55.0 

81.3 
78.1 
75.2 
71.4 
68.5 
66.7 
64.9 
62.5 
60.0 

85.5 
82.0 
78.7 
74.6 
71.4 
69.4 
67.6 
65.0 

89.3 
86.2 
82.0 
78.1 
74.6 
72.5 
70.0 

225   

200   

175  

160   

150   

140   .      .      . 

130     . 

120 

110 

100 

95 

90 

85  

80 





75 

70.  .    . 

65  32  5 

1 1.00* 


Influence  Line  for  M 


120 


LIVE-LOAD    STRESSES 


TABLE  20 

RECIPROCALS  OF  INFLUENCE-LINE  ORDINATES  FOR  M  FOR  GIRDER  BRIDGES 
WITHOUT  FLOOR-BEAMS 

Values  Of  ry 

lit, 
SHORTER  SEGMENT  h 


5 

10 

15 

20 

25 

30 

35 

40 

45 

50 

55 

60 

250 

.204 

.104 

.0707 

.0540 

.0440 

.0374 

.0326 

.0290 

.0262 

.0240 

.0221 

.0207 

225 

.204 

.104 

.0711 

.0544 

.0444 

.0378 

.0330 

.0295 

.0266 

.0244 

.0226 

.0211 

200 

.205 

.105 

.0716 

.0550 

.0450 

.0383 

.0335 

.0300 

.0272 

.0250 

.0232 

.0217 

175 

.206 

.106 

.0723 

.0558 

.0457 

.0390 

.0342 

.0307 

.0279 

.0257 

.0238 

.0224 

160 

.206 

.106 

.0730 

.0562 

.0462 

.0396 

.0348 

.0313 

.0284 

.0263 

.0244 

.0229 

150 

.207 

.107 

.0733 

.0567 

.0466 

.0400 

.0352 

.0317 

.0288 

.0266 

.0248 

.0233 

140 

.207 

.107 

.0738 

.0571 

.0472 

.0405 

.0357 

.0321 

.0293 

.0271 

.0253 

.0238 

130 

.208 

.108 

.0744 

.0577 

.0477 

.0410 

.0363 

.0327 

.0299 

.0277 

.0259 

.0244 

120 

.208 

.108 

.0750 

.0583 

.0483 

.0417 

.0369 

.0333 

.0306 

.0283 

.0265 

.0250 

-4? 

110 

.209 

.109 

.0758 

.0591 

.0491 

.0424 

.0376 

.0341 

.0314 

.0291 

.0273 

.0258 

•** 

d 

100 

.210 

.110 

.0766 

.0600 

.0500 

.0433 

.0386 

.0350 

.0322 

.0300 

.0282 

.0267 

V 

r* 

95 

.211 

.111 

.0772 

.0605 

.0505 

.0438 

.0391 

.0355 

.0327 

.0305 

.0287 

.0272 

a 

90 

.211 

.111 

.0778 

.0611 

.0511 

.0444 

.0397 

.0361 

.0333 

.0311 

.0293 

.0277 

1 

85 

.212 

.112 

.0784 

.0618 

.0517 

.0451 

.0403 

.0368 

.0340 

.0318 

.0299 

.0284 

S3 

80 

.213 

.113 

.0792 

.0625 

.0525 

.0458 

.0411 

.0375 

.0347 

.0325 

.0307 

.0292 

bC 

75 

.213 

.113 

.0800 

.0633 

.0533 

.0466 

.0419 

.0383 

.0356 

.0333 

.0315 

.0300 

§ 

70 

.214 

.114 

.0810 

.0643 

.0543 

.0476 

.0428 

.0393 

.0365 

.0343 

.0325 

.0309 

•J 

65 

.215 

.115 

.0820 

.0654 

.0554 

.0487 

.0440 

.0404 

.0376 

.0353 

.0336 

.0321 

60 

.217 

.117 

.0833 

.0666 

.0567 

.0500 

.0452 

.0417 

.0388 

.0366 

.0348 

.0333 

55 

.218 

.118 

.0848 

.0682 

.0582 

.0515 

.0467 

.0432 

.0404 

.0382 

.0364 

50 

.220 

.120 

.0867 

.0700 

.0600 

.0533 

.0486 

.0450 

.0422 

.0400 

45 

.222 

.122 

.0889 

.0722 

.0622 

.0555 

.0508 

.0472 

.0444 

40 

.225 

.125 

.0917 

.0750 

.0650 

.0583 

.0536 

.0500 

35 

.229 

.129 

.0952 

.0786 

.0686 

.0619 

.0571 

30 

.233 

.133 

.1000 

.0833 

.0733 

.0666 

25 

.240 

.140 

.1066 

.0900 

.0800 

20 

.250 

.150 

.1166 

.1000 

15 

.267 

.167 

.1333 

10 

.300 

.200 

5 

.400 

LIVE-LOAD    STRESSES 


121 


TABLE  20.—  Continued 

RECIPROCALS  OF  INFLUENCE-LINE  ORDINATES  FOR  M  FOR  GIRDER  BRIDGES 
WITHOUT  FLOOR-BEAMS 

Values  of 


SHORTER  SEGMENT 


65 

70 

75 

80 

85 

90 

95 

100 

110 

120 

130 

140 

250 
225 
200 
175 
160 
150 
140 
130 
120 
110 
100 
95 
90 
85 
80 
75 
70 
65 

.0194 
.0198 
.0204 
.0212 
.0217 
.0221 
.0225 
.0231 
.0237 
.0245 
.0254 
.0259 
.0265 
.0272 
.0279 
.0287 
.0296 
.0307 

.0183 
.0188 
.0193 
.0200 
.0206 
.0210 
.0214 
.0220 
.0226 
.0234 
.0243 
.0248 
.0254 
.0261 
.0268 
.0276 
.0286 

.0174 
.0178 
.0183 
.0191 
.0196 
.0200 
.0204 
.0210 
.0216 
.0224 
.0233 
.0238 
.0244 
.0251 
.0258 
.0286 

.0165 
.0170 
.0175 
.0182 
.0188 
.0192 
.0196 
.0202 
.0208 
.0216 
.0225 
.0230 
.0236 
.0243 
.0250 

.0158 
.0162 
.0168 
.0175 
.0180 
.0184 
.0189 
.0194 
.0201 
.0208 
.0217 
.0223 
.0229 
.0235 

.0151 
.0156 
.0161 
.0168 
.0174 
.0178 
.0183 
.0188 
.0194 
.0202 
.0211 
.0216 
0222 

.0145 
.0150 
.0155 
.0162 
.0168 
.0172 
.0177 
.0182 
.0188 
.0196 
.0205 
.0211 

.0140 
.0144 
.0150 
.0157 
.0162 
.0167 
.0171 
.0177 
.0183 
.0191 
.0200 

.0131 
.0136 
.0141 
.0148 
.0154 
.0158 
.0162 
.0168 
.0174 
.0182 

.0123 
.0128 
.0133 
.0140 
.0146 
.0150 
.0154 
.0160 
.0167 

.0117 
.0122 
.0127 
.0134 
.0140 
.0144 
.0148 
.0154 

.0112 
.0116 
.0122 
.0128 
.0134 
.0138 
.0143 









• 

Influence  Line  for  M 


122 


LIVE-LOAD    STRESSES 


TABLE  21 

BENDING  MOMENTS  IN  BEAMS  DUE  TO  UNIFORM  LOAD  OP  1  POUND  PER 

LINEAL  FOOT 

Values  in  Foot-pounds 

Values  equal  -^  =  Area  of  Influence  Line  for  M 
SHORTER  SEGMENT  h 


5 

10 

15 

20 

25 

30 

35 

40 

4b 

50 

55 

60 

250 

625 

1250 

1875 

2500 

3125 

3750 

4375 

5000 

5625 

6250 

6875 

7500 

225 

562.5 

1125 

1687.5 

2250 

2812.5 

3375 

3937.5 

4500 

5062.5 

5625 

6187.5 

6750 

200 

500 

1000 

1500 

2000 

2500 

3000 

3500 

4000 

4500 

5000 

5500 

6000 

175 

437.5 

875 

1312.5 

1750 

2187.5 

2625 

3062.5 

3500 

3937.5 

4375 

4812.5 

5250 

160 

400 

800 

1200 

1600 

2000 

2400 

2800 

3200 

3600 

4000 

4400 

4800 

150 

375 

750 

1125 

1500 

1875 

2250 

2625 

3000 

3375 

3750 

4125 

4500 

140 

350 

700 

1050 

1400 

1750 

2100 

2450 

2800 

3150 

3500 

3850 

4200 

130 

325 

650 

975 

1300 

1625 

1950 

2275 

2600 

2925 

3250 

3575 

3900 

120 

300 

600 

900 

1200 

1500 

1800 

2100 

2400 

2700 

3000 

3300 

3600 

^ 

110 

275 

550 

825 

1100 

1375 

1650 

1925 

2200 

2475 

2750 

3025 

3300 

+a 

100 

250 

500 

750 

1000 

1250 

1500 

1750 

2000 

2250 

2500 

2750 

3000 

g 

95 

237.5 

475 

712.5 

950 

1187.5 

1425 

1662.5 

1900 

2137.5 

2375 

2612.5 

2850 

g 

90 

225 

450 

675 

900 

1125 

1350 

1575 

1800 

2025 

2250 

2475 

2700 

o3 

85 

212.5 

425 

637.5 

850 

1062.5 

1275 

1487.5 

1700 

1912.5 

2125 

2337.5 

2550 

r/j 

80 

200 

400 

600 

800 

1000 

1200 

1400 

1600 

1800 

2000 

2200 

2400 

I 

75 

187.5 

375 

562.5 

750 

937.5 

1125 

1312.5 

1500 

1687.5 

1875 

2062.5 

2250 

a 

70 

175 

350 

525 

700 

875 

1050 

1225 

1400 

1575 

1750 

1925 

2100 

2 

65 

162.5 

325 

487.5 

650 

812.5 

975 

1137.5 

1300 

1462.5 

1625 

1787.5 

1950 

60 

150 

300 

450 

600 

750 

900 

1050 

1200 

1350 

1500 

1650 

1800 

55 

137.5 

275 

412.5 

550 

687.5 

825 

962.5 

1100 

1237.5 

1375 

1512.5 

.  .  .  . 

50 

125 

250 

375 

500 

625 

750 

875 

1000 

1125 

1250 



45 

112.5 

225 

337.5 

450 

562.5 

675 

787.5 

900 

1012.5 

40 

100 

200 

300 

400 

500 

600 

700 

800 

35 

87.5 

175 

262.5 

350 

437.5 

525 

612.5 

30 

75.0 

150 

225 

300 

375 

450 

25 

62.5 

125 

187.5 

250 

312.5 

20 

50.0 

100 

150 

200 

15 

37.5 

75 

112.5 

10 

25.0 

50 

5 

12.5 

LIVE-LOAD    STRESSES 


123 


TABLE  21.— Continued 

BENDING    MOMENTS  IN  BEAMS  DUE  TO  UNIFORM  LOAD  OF  1  POUND  PER 

LINEAL  FOOT 

Values  in  Foot-pounds 

Values  equal  ^  =  Area  of  Influence  Line  for  M 
SHORTER  SEGMENT  h 


65 

70 

75 

80 

85 

90 

95 

100 

110 

120 

130 

140 

250 
225 
200 
175 
160 
150 
140 
130 
120 
110 
100 
95 
90 
85 
80 
75 
70 
65 

8125 
7312.5 
6500 
5687.5 
5200 
4875 
4550 
4225 
3900 
3575 
3250 
3087.5 
2925 
2762.5 
2600 
2437.5 
2275 
2112.5 

8750 
7875 
7000 
6125 
5600 
5250 
4900 
4550 
4200 
3850 
3500 
3325 
3150 
2975 
2800 
2625 
2450 

9375 
8437.5 
7500 
6562.5 
6000 
5625 
5250 
4875 
4500 
4125 
3750 
3562.5 
3375 
3187.5 
3000 
2812.5 

10000 
9000 
8000 
7000 
6400 
6000 
5600 
5200 
4800 
4400 
4000 
3800 
3600 
3400 
3200 

10625. 
9562.5 
8500 
7437.5 
6800 
6375 
5950 
5525 
5100 
4675 
4250 
4037.5 
3825 
3612.5 

11250 
10125 
9000 
7875 
7200 
6750 
6300 
5850 
5400 
4950 
4500 
4275 
4050 

11875 
10687.5 
9500 
8312.5 
7600 
7125 
6650 
6175 
5700 
5225 
4750 
4512.5 

12500 
11250 
10000 
8750 
8000 
7500 
7000 
6500 
6000 
5500 
5000 

13750 
12375 
11000 
9625 
8800 
8250 
7700 
7150 
6600 
6050 

15000 
13500 
12000 
10500 
9600 
9000 
8400 
7800 
7200 

16250 
14625 
13000 
11375 
10400 
9750 
9100 
8450 

17500 
15750 
14000 
12225 
11200 
10500 
9800 

1  Pound  per  Lineal  Foot 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 

AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  SO  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $I.OO  ON  THE  SEVENTH  DAY 
OVERDUE. 


MAR  161934 


LD  21-100m-7,'33 


337856 


- 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


